At 10 weeks, his little bladder can't hold it very long - about three or four hours, max. Is he crate-trained? That would help a lot if he isn't already.
On your own time, before the time when he starts whining, wake him up and take him out to go potty, then put him back in the crate. I suggest putting a few kibble-stuffed Kongs in his crate after this walk to keep him occupied. If he still whines afterwards, don't talk to him, let him out, don't pay him any attention at all, or he'll associate whining with that. I know it's hard, but it should go away in a few weeks.