Preston came from a goal down to beat Nottingham Forest on the final day of the season on Saturday.
James Garner opened the scoring for the hosts before goals from Bayliss and Liam Lindsay earned the Lilywhites all three points.
The 22-year-old has backed McAvoy for the top job after the Scot picked up 17 points form a possible 24 in his eight games in charge.
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Preston North End defender says previous defeat inspired them to victory over Luton Town this time around
He said: “We went from having the worst form to now having one of the best in the league towards the end of the season.
“It’s been brilliant – what’s happened since Frankie’s been in charge is brilliant for us all. We’ve got to leave appointing the new manager to the powers that be but I think Frankie has done more than enough to stake his claim for the job.
“He’s been brilliant since he’s come in and all the lads love him so we’ll have to see what happens in the summer.”
Bayliss came in for his second Championship start for PNE, his previous coming in the reverse fixture.
He has had just a handful of minutes appearing from the bench in recent weeks but knew that when his chance came he would have to take it.
He said: “We started the first half quite well and then they got on top and we struggled to deal with it. They got their goal and we kind of hung in there until half-time.
“We made a few changes at half- time and it worked, it was really good for us. It’s really good to get a goal and finish the season on a high.
“Coming back from 1-0 down is really good for us.
“I just wanted to show what I could do in those areas. I was out on the right-hand side at some points today and I was trying to get closer to Ched Evans and that’s how I got my goal. It came as a bit of a surprise to be starting because the lads had been doing really well so I wasn’t really expecting it. You’ve got to take those chances when they come along.
“I’ll always come in and work – your work rate and work ethic is what will get you into teams. That’s second nature for me.”