Though Accies are not officially out of the top flight just yet, they need at least a nine-goal win over Sunday’s opponents Kilmarnock in their final game to avoid finishing last in the division. For County, meanwhile, they just require a point from their trip to Motherwell to secure safety for another season.
The visitors looked to be on their way to getting the result they required when Scott McMann fired them into a first-half lead, but Blair Spittal equalised within two minutes and a 70th-minute winner from substitute Charlie Lakin sealed their fate.
Rice previously worked as an assistant to Hughes at Falkirk, Hibs and Inverness CT and after the match his former boss praised the visitors for their determination.
Hughes told BBC Sportsound: “I have to pay commiserations to Hamilton. Brian Rice is a really good mate of mine and they've absolutely different class in the Scottish Premiership for a number of years. They've got something about them, that never-say-die attitude, that I wish I could get here on a more regular basis.
“I wish them a speedy return to the top flight and I'm looking for a favour from them on Sunday against Kilmarnock. I'll be on the phone making sure he's got his team up and ready to go.
“It's still all to play for. We'll need to be at it. We're playing against a right good Motherwell team. Graham Alexander has done a great job and we need to be at our best.”
Rice praised his players for their effort despite the defeat as the visitors created a number of chances that, with a bit of good fortune, would have saw them live to fight another day.
He said: “We're absolutely devastated. I'm devastated for everybody connected with Hamilton.
“I thought we played very, very well. We created numerous chances, we just couldn't get over the line.
“The players gave it their all tonight. I asked them to play with a freedom, with no fear, and you saw a team playing exceptionally well. It just wasn't enough.”