To link an individual to an employer there is a right way and a wrong way.
If they are changing companies NEVER edit the existing relationship to change the company name. Simply enter an end date into the existing relationship
Then add the new relationship. The only hard part about it might be finding the correct organization - because sometimes there are several with similar names. If they want to be linked to a member organization - you need to find the organization that has the membership.
The best way is to search for organizations, and find the right one.
For example there are many Wells Fargo records. Some of these convey member benefits, some don't.
Choose the most relevant organization that has member benefits - as shown by the receives benefits and member flag being checked
Then you can add the relationship, by using the "+" sign at the right hand end of the relationship tab
Just use the form to add the relationship. REMEMBER to check the primary key. Use the relationship type "Employee"