The 24-year-old, who came through Everton's academy, has spent time in Scotland, where Latics manager Lee Johnson spent part of his own playing career, before returning to England to ply his trade in non-league football.
Speaking to the club's official website, Johnson said: "JP is an interesting one, he is someone I have followed for a long time; I have always thought he was a really good player.
"It may appear to be an unfashionable signing, but sometimes as a manager you have to weigh up the value, age, pedigree and talent and take a chance. This is such an occasion.
"He is certainly hungry, and with the pedigree he has got he needs to play with players that can fit with a similar style. It's up to him now to show is worthy of being a League One player - if not above. The lads have really taken to him as he has something different and he can certainly add to our game, whether it be from the bench or starting."
Kissock himself is grateful to Oldham for the chance to test himself in the third tier.
He added: "It's an unbelievable opportunity that the gaffer has offered me.
"To be back in the Football League after a few years in non-league is something I have worked hard for. I am delighted to be given this chance and I am really looking forward to being part of the squad travelling to Swindon for Saturday's game."