I guess I tend to favor on the side of "ask permission first, if denied, then find another way to do it or re-evaluate." Perhaps that's the anxiety speaking, though. Exploring is a hobby. I certainly don't want to go to jail over it. The main method I've used in the past is to email or call property owners/caretakers or town officials and state that I'm doing research. Research is great. You can take pictures for research, you can poke around, and you can pretty much be researching ANYTHING. If it's a historic site (and most are) I do a quick google search and come up with a relative "thesis." Once I had a friend pose as a professor. I always try to do "research" that is non-controversial and is parallel or in the best interest of whomever has the control. For example, I happen to be Jewish, but when I went to explore Holyland, USA, i avidly posed as someone who "deeply believed in the worth of such a location and the glory it had in it's youth". Yeah....in reality...not so much. You also can really look trustworthy if you ask for or let people give you a guided tour, as long as you also ask at the end that they "leave you alone so you can form unbiased opinions." It can make them feel safer that you won't vandalize.
Some other excuses I've cited:
-Religious ceremonies
-Car broke down
-Looking for an address (must be near roads/civilization)
-Geocaching (Even if there aren't caches nearby, you can say there are and that you're just looking for them. Helps to have your phone out.)
-Trying to get cell signal
-Needed to pee