To clarify a little further, there need not be any contract between the federal or state government and any hotel. The government entity announces the rates that it is willing to pay and each hotel decides whether or not to offer rooms at that rate. The federal government is large enough (maybe too large, but that's for another discussion) that hotels pretty routinely include the GSA rates in their standard room rates, which may or may not be available for any given date. For states, the situation may be different. A hotel in Chicago may have standard rates for Illinois state employees, but probably doesn't include other states' rates on its website. That doesn't mean that it won't offer rooms at the Tennessee rate, but you'll have to speak to someone at the hotel to get a decision. "Extras" beyond a room may also be available, but the hotel may not want to list all of the combinations that would accumulate.

So, call the hotel and speak with someone with enough authority to make a decision.