GSA has a clause called the Most Favored Customer (MFC) clause. It requires the contractor to offer the government at least the best price it offered to a previous customer. GSA uses this clause for its schedules. There is no reason your agency can't adopt this clause for its own contracts; one agency doesn't have a monopoly on clauses. That being said, you may get fewer contractors than you wanted by including it in your solicitation. Still, it may be worth it in a sealed bid or lowest price technically acceptable (LPTA) situation.
Of course, not everyone is happy about MFC clauses, and I can understand why. Give it some though. Do you know people who have used this clause in non-GSA contracts?