^^^agree with weaver. The hind milk should Austin him for longer too.
In relation to the docs recommendation to get on a "schedule": I think it's far too early for that just yet - especially considering Baby was early. IMO.
If it were me, I'd be following Baby's cues and lead and letting them set the pace right now. I was on a 2hr feeding (round the clock) at that stage and only moved to having a 3/ more hr stretch at night time later. If Baby is giving hungry cues and looking for a BF, I'd give it.
Well done for a brilliant job so far as he sounds like you are doing a great job for him!!