• Salesforce has introduced LogoutEventStream in Summer’18 release, to get the User logout events.
• It is a beta feature and to make it available in your org, you need to contact Salesforce Support.
• After it will available in your Org, you can enable LogoutEventStream.
• After enabled LogoutEventStream, Salesforce publishes logout events when users log out from the UI.
• You can add an Apex trigger to subscribe to those events, where you can implement our custom logic during logout.

Enable Logout Events Stream :
Go to Setup | Event Manager (Enter Event Manager in Quick Find) | Click on Logout Event link and click on Update Event on right coroner button and select Enable Streaming.

After enable Logout Events Stream, I have created an Apex trigger on LogoutEventStream object, where I’m saving the User Logout Event information in a Custom Object.

Custom Object:

Apex Trigger:

trigger LogoutEventTrigger on LogoutEventStream (after insert) {
    
    List<LogoutEventInfo__c> leInfoList = new List<LogoutEventInfo__c>();
    
    for(LogoutEventStream les : Trigger.new){
        LogoutEventInfo__c leInfo = new LogoutEventInfo__c();
        leInfo.EventIdentifier__c = les.EventIdentifier;
        leInfo.UserId__c = les.UserId;
        leInfo.Username__c = les.Username;
        leInfo.EventDate__c = les.EventDate;
        leInfo.RelatedEventIdentifier__c = les.RelatedEventIdentifier;
        leInfo.SessionKey__c = les.SessionKey;
        leInfo.LoginKey__c = les.LoginKey;
        leInfo.ReplayId__c = les.ReplayId;
        leInfo.SessionLevel__c = les.SessionLevel;
        leInfo.SourceIp__c = les.SourceIp;
        leInfoList.add(leInfo);
    }
    Insert leInfoList;
}

Now, you can Logout and Login again to check the Logout Event information in Custom Object.