Killie suffered their eighth loss in nine matches in all competitions on Saturday to fall second bottom of the Scottish Premier League, remaining just a point above Falkirk.
Jimmy Calderwood's men must start winning soon if they are to avoid their 17-year stay in the top flight coming to an end.
It is a near-identical situation to that they found themselves in last season, when they also lost their first post-split fixture at New Douglas Park before embarking on a run that yielded 10 points from their final four games.
Despite Saturday's dispiriting 3-0 reverse, club captain Bryson sees no reason why they cannot do the same again.
"We've got quite good players in the squad that can do that," said the midfielder, who was relieved to learn neither Falkirk or fellow strugglers St Mirren won on Saturday.
"Last season, in the split, we lost to Hamilton and then went on and won three of our last four.
"We need to start picking up points soon or we could be in trouble."
That does not just apply on the field, with Killie in serious debt and relegation likely to see a fire sale - or worse - at Rugby Park.
Bryson said: "Obviously, it's no secret that the club's in a lot of debt and going down isn't going to help that.
"We just try to keep playing football; we've got a job to do here to try to keep the team in the SPL.
"If we can do that then that's our job done and hopefully all will be okay."
The 23-year-old admits the prospect of relegation makes him fear for his own future.
"It always does when a team goes down and they look to offload players," he said.
"But we try not to think about that and just try to do our job on a Saturday."
For much of yesterday's game, they did just that.
But once Dougie Imrie gave Hamilton a 72nd-minute lead, they simply collapsed, conceding further goals to Flavio Paixao and Joel Thomas.
Having only scored four times away from home all season - by far the worst return in British league football - it was no surprise to see the visitors' heads drop once they fell behind.
"I don't know what the problem is," said Bryson.
"We've lost 10 games away from home 1-0, so we've not been coming back.
"It looks like the first goal's going to be crucial for us. If we get it, we can hang on."
Unfortunately for Killie, their next game is also away from home against one of their two relegation rivals.
Bryson said: "We've got a big game next week against St Mirren and we just need to get it right."
Accies striker Imrie knows just what it feels like to be relegated, having gone down with Inverness last season.
The 26-year-old, who moved to New Douglas Park on transfer deadline day two months ago, had feared another dogfight when he first joined Hamilton.
But after helping his new club net their fifth win in seven games, he believes they are unlucky not to have been in the shake-up for the top six.
"At the start of the season - I wasn't here - they had a bad run," said Imrie, who scored his first Accies goal yesterday.
"Alec Neil was out until November. Mark McLaughlin and James McArthur were suspended for the first two games.
"If those players had all started the season, who knows where we could've been?"