Hi Zoom,

Can you try this code.  I hope this will solve.

trigger AutoCreateSubsServOnContrOv on Contract_Overview__c (after insert, after update) 
{
   List<Subs_Serviced_On_Contract__c> subs = new List<Subs_Serviced_On_Contract__c>();

   // get the full list of sub account names for all records being processed by the trigger

   
   Map<Id, List<String>> ContractToSubsAccNames =  new Map<Id, List<String>>();
   Map<Id, Contract_Overview__c> contractMap = new Map<Id, Contract_Overview__c>();
   for (Contract_Overview__c newCont : Trigger.New) 
   {
      List<String> subAccNames=new List<String>();
	  if (newCont.Subsidiaries_On_Contract__c != '[]') 
      {
         // split out the multi-select picklist using a comma delimiter
         System.debug('Subsidiaries_On_Contract__c ' + newCont.Subsidiaries_On_Contract__c);
		          String temp = newCont.Subsidiaries_On_Contract__c;
         temp = temp.replace(']','');
         temp = temp.replace('[','');
         subAccNames = temp.split(',');
	  }
      ContractToSubsAccNames.put(newcont.Id, subAccNames);
      contractMap.put(newcont.Id, newcont);
   }
   System.debug('*******subAccNames List Size::::::'+ subAccNames.size());
   
   //get the map for contract and child object Subs_Serviced_On_Contract__c
   Map<Id, List<Subs_Serviced_On_Contract__c>>  contToSubsMap = new  Map<Id, List<Subs_Serviced_On_Contract__c>>();
   // Replace Subs_Serviced_On_Contracts__r value in below query  with child relationship name
   for(Contract_Overview__c co : [Select Id, (Select  Name,Contract_Overview__c,Account__c,Subsidiaries_and_Brands__c ,Contract_and_Sub__c,Contract_Start_Date__c = newContract.Contract_Start_Date__c, Contract_End_Date__c,logo_Usage_Allowed__c from Subs_Serviced_On_Contracts__r) from Contract_Overview__c where id in :contractMap.keySet()])
	{
        contToSubsMap.put(co.id, co.Subs_Serviced_On_Contracts__r);
	}

   // get the ids for all sub accounts and store in a map keyed by name  
   Map<String, Id> subAccIdsByName=new Map<String, Id>();
   for (SubsidiariesAndBrands__c subAcc : [select id, Name from SubsidiariesAndBrands__c where Name in :ContractToSubsAccNames.Values()]) 
    {
              subAccIdsByName.put(subAcc.Name, subAcc.id);
    }
   System.debug('SubsidiariesAndBrands size ************ ' + subAccIdsByName.size());
   
   //For each position processed by the trigger, add a new  

   //Subs_Serviced_On_Contract record for the specified Subsidiaries_On_Contract__c.  

   //Note that Trigger.New is a list of all the new positions  

   //that are being created.  

   for (Id ConId : contractMap.ketSet()) 
   {
      
	  if(contToSubsMap.containsKey(ConId.id) && contToSubsMap.get(ConId.id).size() > 0)
	   {
		  // When Update occurs on contract this loop will excecute
		  for(Subs_Serviced_On_Contract__c sscoupdate : contToSubsMap.get(ConId.id))
		   {
			  // contract has already child Subs_Serviced_On_Contract__c records we need to update with update contract values
			  String ssocName = sscoupdate.Name.normalizeSpace();
			  if(subAccIdsByName.containsKey(ssocName))
			   {
				  // Subs_Serviced_On_Contract__c record matches then update
				  sscoupdate.Contract_Overview__c = contractMap.get(ConId.id).Id;
                  sscoupdate.Account__c = contractMap.get(ConId.id).Account__c;
                  sscoupdate.Subsidiaries_and_Brands__c = subAccIdsByName.get(subsoncontract);
				  // GET THE SUB ACCOUNT ID BASED ON NAME
                  sscoupdate.Contract_and_Sub__c = newContract.Name + '~' + ssocName;
                  sscoupdate.Contract_Start_Date__c = contractMap.get(ConId.id).Contract_Start_Date__c;
                  sscoupdate.Contract_End_Date__c = contractMap.get(ConId.id).Contract_End_Date__c;
                  sscoupdate.Logo_Usage_Allowed__c = 'Yes';

				  subs.add(sscoupdate);

			   }else{
				   // Subs_Serviced_On_Contract__c record not matched then insert
				   Subs_Serviced_On_Contract__c ssocnew = new Subs_Serviced_On_Contract__c(
                    Name = subsoncontract,
                    Contract_Overview__c = contractMap.get(ConId.id).Id,
                    Account__c = contractMap.get(ConId.id)t.Account__c,
                    Subsidiaries_and_Brands__c = subAccIdsByName.get(subsoncontract),  // GET THE SUB ACCOUNT ID BASED ON NAME
                    Contract_and_Sub__c = contractMap.get(ConId.id).Name + '~' + subsoncontract,
                    Contract_Start_Date__c = contractMap.get(ConId.id).Contract_Start_Date__c,
                    Contract_End_Date__c = contractMap.get(ConId.id).Contract_End_Date__c,
                    Logo_Usage_Allowed__c = 'Yes');

					subs.add(ssocnew);
			   }
		   }
	   }else if (contToSubsMap.containsKey(ConId.id) && contToSubsMap.get(ConId.id).size() == 0)
	     {
		    // When first insert  occurs on contract and no child records this loop will excecute
		   for(String subsoncontract : ContractToSubsAccNames.get(ConId.id))
		   {
			   subsoncontract = subsoncontract.normalizeSpace();
               Subs_Serviced_On_Contract__c ssoc = new Subs_Serviced_On_Contract__c(
                    Name = subsoncontract,
                    Contract_Overview__c = newContract.Id,
                    Account__c = newContract.Account__c,
                    Subsidiaries_and_Brands__c = subAccIdsByName.get(subsoncontract), // GET THE SUB ACCOUNT ID BASED ON NAME
                    Contract_and_Sub__c = contractMap.get(ConId.id).Name + '~' + subsoncontract,
                    Contract_Start_Date__c = newContract.Contract_Start_Date__c,
                    Contract_End_Date__c = newContract.Contract_End_Date__c,
                    Logo_Usage_Allowed__c = 'Yes');

              subs.add(ssoc);
		   }
		 }
   }
   if(!sub.isEmpty())
	   upsert sub;
}