Thanks a lot for the advice. Would you be able to help me a bit further and take a look at the contact trigger and handler class and let me know how do I modify the trigger to get this done.
​​​​​​​This issue is not limited to delete, I need the field to update when a new contact record is added as well.
One more thing to note here is that there is another cusotm object involed Correspondence Preference with master detail relationship, contact being the master. So until a corresponding record on Correspondence Preference object is created, I do not expect anything to change on communication content object. Would be glad to walk you through the whole functionality if that helps.
Here is the trigger:

trigger ContactTrigger on Contact (after insert, after update,after delete,after undelete) { 
    
    if(Trigger.isAfter) {
            if(Trigger.isUpdate) {
            ContactTriggerHandler.onAfterUpdate(Trigger.new,Trigger.oldMap);
            }
            
            if(Trigger.isInsert){
             ContactTriggerHandler.OnAfterInsert(Trigger.new);
            }
            
            if(Trigger.IsDelete){
                ContactTriggerHandler.OnAfterDelete(Trigger.old);
            }
            
            if(Trigger.IsUnDelete){
                ContactTriggerHandler.OnUndelete(Trigger.new);
            }
    }
     
}

Here is the handler apex class:

public class ContactTriggerHandler {
    
    public static void onAfterUpdate(List<Contact> newContacts, Map<id,Contact> oldContacts) {
            updateSortedCorrespondencePreferneceData(newContacts,oldContacts);
            updateSalutationOnCommunicationContent(newContacts,oldContacts);             
            id cpCapitalPartnerRT = fetchRecordTypeId('Account','CP Capital Partner');
            List<Contact> newList = new List<Contact>();
            
            set<Id> oldAndNewAccountIds = new set<id>();
            for(Contact con : newContacts) {
                oldAndNewAccountIds.add(con.accountId);
                oldAndNewAccountIds.add(oldContacts.get(con.id).accountId);
            }
            
            Map<Id,Account> accountDetails = new Map<Id,Account>([Select Id,RecordTypeId from Account where Id in :oldAndNewAccountIds]);
            for(Contact con : newContacts) {
                    if(accountDetails !=NULL && con!= Null && con.accountId != oldContacts.get(con.Id).accountId  ) { 
                        if(  accountDetails.containsKey(con.accountId) && (accountDetails.get(con.accountId).recordTypeId == cpCapitalPartnerRT || accountDetails.get(oldContacts.get(con.Id).accountId).recordTypeId == cpCapitalPartnerRT) ) {
                            newList.add(con);
                        }

                    } else if(accountDetails !=NULL && con!= Null && accountDetails.get(con.accountId).recordTypeId == cpCapitalPartnerRT ) { 
                        newList.add(con);
                    }
            }        
           
            Email_InformationClass.getEmailTableData(newList,oldContacts); 
            updateCPIntralinksAccess(newContacts,oldContacts);  
    }
    
    public static void OnAfterDelete(contact[] oldList){
        
        updateCPIntralinksAccess(oldList, null);
    }
    
     public static void OnAfterInsert(contact[] newList){
        
        updateCPIntralinksAccess(newList, null);
    }
     
    public static void OnUndelete(contact[] newList){ 
         
         updateCPIntralinksAccess(newList,null);
    }
	private static void updateSalutationOnCommunicationContent(List<Contact> newContacts, Map<id,Contact> oldContacts){
        set<id> contactsToProcess = new set<id>();
        for(Contact con : newContacts) {
            if((String.isNotBlank(con.email) && !con.email.contains('@generalatlantic.com'))  && ((con.Seniority_Rank__c != oldContacts.get(con.Id).Seniority_Rank__c) ||
                                                                                                  (con.FirstName != oldContacts.get(con.id).FirstName)) || (con.International_Salutation_Type__c!= oldContacts.get(con.Id).International_Salutation_Type__c || String.isBlank(oldContacts.get(con.Id).email))
              )
            {
                
                        contactsToProcess.add(con.id);
            }
        }
        if(contactsToProcess.size() > 0) {
            List<Correspondence_Preference__c> corrPrefToProcess = new List<Correspondence_Preference__c>([Select Id, Company__c, RecordType.Name from Correspondence_Preference__c where Recipient_Type__c = 'To' AND Contact__c In :contactsToProcess]);
            set<string> cpNames = new Set<String>();
            set<string> accountNames = new Set<String>();
            for(Correspondence_Preference__c cp : corrPrefToProcess) {
                cpNames.add(cp.RecordType.Name);
                accountNames.add(cp.Company__c);
            }
            List<Communications_Content__c> ccList = new List<Communications_Content__c>([Select Id,Salutation__c,Company__c,Correspondence_Preference__c from Communications_Content__c where Company__c IN :accountNames AND Correspondence_Preference__c IN :cpNames]);
            CommunicationContentTriggerHandler handler = new CommunicationContentTriggerHandler(true,200,true);
            handler.updateSalutationField(null,ccList); 
        }
    } 
    private static void updateSortedCorrespondencePreferneceData(List<Contact> newContacts, Map<id,Contact> oldContacts) {
        set<id> contactsToProcess = new set<id>();
        for(Contact con : newContacts) {
            if((con.Seniority_Rank__c != oldContacts.get(con.Id).Seniority_Rank__c) ||
               (con.Email != oldContacts.get(con.id).Email)
              ) {
                  contactsToProcess.add(con.id);
              }
        }
        
        if(contactsToProcess.size() > 0) {
            List<Correspondence_Preference__c> corrPrefToProcess = new List<Correspondence_Preference__c>([Select Id, Commitment__c, RecordTypeId from Correspondence_Preference__c where Contact__c In :contactsToProcess]);
            
            CorrespondenceTriggerHandler.getCommRecTypeData(corrPrefToProcess,null,false);
        }
    }
    private static id fetchRecordTypeId(string sObj, string rt) {
        System.debug('fetch params>>'+sObj + '<<<rt>>>'+rt);
        String recordtypeId = Schema.getGlobalDescribe().get(sObj).getDescribe().getRecordTypeInfosByName().get(rt).getRecordTypeId(); 
        return Schema.getGlobalDescribe().get(sObj).getDescribe().getRecordTypeInfosByName().get(rt).getRecordTypeId();
        
    }
    public static void updateCPIntralinksAccess(List<Contact> newContacts, Map<id,Contact> OldMap){
        
        Map<ID,List<contact>> mapConWithAcc = new Map<ID,List<contact>>();
        List<account> accList = new List<account>();
        String temp= '';
        Map<Id,String> accUpdateMap = new Map<Id,String>();
        Set<ID> accid = new Set<ID>();
        
        for(contact con: newContacts){
            
            if(oldMap == null || oldMap.get(con.id).CP_Intralinks_Access__c != con.CP_Intralinks_Access__c ){ 
                accid.add(con.accountid); 
                
            }
        }
        
        for(Account acc : [Select id,CP_Intralinks_Access__c,(Select id,CP_Intralinks_Access__c,accountid from contacts) from account where id in :accid]) {
            
            set<string> val = new set<string>();
            
            String finalVal = '' ;
            
            for(Contact ct : acc.contacts) {
                
                if(String.isNOtBlank(ct.CP_Intralinks_Access__c)) {
                    
                    val.addAll(ct.CP_Intralinks_Access__c.split(';')); 
                    
                }
                
                
                
            }
            
            List<String> valList = new List<String>();
            
            valList.addAll(val);
            
            if(valList.size() > 0) {
                
                finalVal = String.join(valList,',');
                
            }
            
            acc.CP_Intralinks_Access__c = finalVal ;
            
            accList.add(acc);
            
        }
        
        
        if(accList.size() > 0) {
            update accList ;
        }
        
        
    }
}