If you’re choosing hospital to deliver, should start going for hospital tour once you’re pregnant to find out which hospital you want to deliver at, same as when you start to look for gynae.
If you’re following gynae, then the hospital is probably fixed according to where he or she can deliver at. If that’s the case, I would prefer to start around 2nd trimester as it may be tough and tiring to carry a huge belly to walk around in 3rd trimester.