It would surely be in the gov't's interest to off-load the management of rent supplement to an (private?) agency tasked with finding the cheapest rents. They could be incentivized to do this, and it provide income to rental agencies as well as saving the state money.
The thing about rent suppliment is that the person choses where they want to live. SW has no roll to play in deciding locations, properties etc. Obviously, the person is going to pick the best accommodation that SW is willing to pay for. Hiring an agency would just add a layer of unnecessary administration. Cutting the allowance is the easiest way to stop the waste. If you did hire outside help, it would be to survey what the average accommodation costs are for working people earning minimum wage and use it as a threshold.