I suggest you create a program that allows an individual to receive an incentive award relative to the savings they create.
If an employee researches is able to officially travel with an airline that is more advantageous to the government than the contract carrier, they can submit submit what is necessary. Since there is already a government contract, it would be reasonably easy to verify.
Likewise, any employee willing and able (based on availability)to stay in a hotel that is LESS than per diem rate, they could receive a percentage of the savings.
In addition, if the government is paying to move the individual and there are more than one possible options (They move themselves or they hire movers) they can get a percentage of those savings.
If this was integrated into the travel voucher submission process, it would be efficient to implement and immediately incentivise employees.