I walked down to Nara Park around 9a.m and there were way less people there than in the afternoon. It’s open 24 hours and no entry gates so you can pretty much come and go as you please. Some of the shrines are open certain times so better check those if you’re planning on going inside. The deer seem to be around at all times and in all parts of Nara Park so you don’t need to worry about not seeing them, although they seem to be out in full force when there’s most people around because they know they’re going to fed.
If you enter from the northern end, there’s a lot of trees with some nice vivid colours, and less people around. You can get to the main “street” or bit, which is where you can buy massive amounts of ice cream (nothing beats Japanese soft cream!), souvenirs you will probably throw away within a year, and deer food. If you go later in the morning or after lunch, stay away from that bit unless you want to be entertained by tourists trying to escape from deer. Once they know you have deer food, they will follow you around like a Donald Trump groupy, try to get into your bag or wherever you’ve tried to hide it. All good fun but some people will freak out.
Have a walk around the edges of Nara park and get away from the parts you see listed on the Nara Park guides, it’s not so crowded and you can still have fun hanging out with the deer. It’s a much more relaxing way to see it then getting trampled by the hoards of tourists who tend to congregate around the main drag of Nara Park.
Nara is easy to get to and you could actually quite easily do Nara Park as a day trip from Osaka or Kyoto. Hire a car, it’s much easier to get around and car parks are relatively easy to find. You can find some on Toucan’s Nara Map.