I think that they'll really, really want to, but his position is untenable and I think they'll know that. I think Jim will too.
It's a real shame, everything about him as a character suggests he should have the qualities to be a really good manager. A strong work ethic, good leadership (St Mirren cup winning captain etc), doesn't suffer fools, intelligent and articulate but grounded. I can see why the board would think that they're so close and want him to keep learning. However, his tactical ineptitude (immaturity, even) isn't going to be overcome overnight and I suspect his understanding and patience will have already boiled over with the players making the dressing room untenable too.
I expect we'll see him doing a good job elsewhere soon. We won't see any of the 3-1-4-2s or the other nonsense setups when there is no pressure on him to produce entertainment elsewhere and he'll become a cautious, disciplined manager ala McInnes and be a good fit for someone.