Hi Stephen, I remember the chipmunks on top of Angel's Landing in Zion! They were everywhere. Is that where you were? Or were you rock climbing?
I'm no expert on getting safely in close contact with wildlife. I have made two seperate trips specifically to see tigers in the wild and spent several days in each location... NOTHING. There are no guarantees where nature is concerned. The best advice I can give is (a) make the effort, research where and when to go and be willing to go to some fairly remote places (depending on what you want to see); (b) be really patient, most animals have huge territories; (c) Time of Day matters usually - know when the species you are interested in are most active; (d) If you are really serious, have a naturalist guide with you who knows how to spot things; (e) be really quiet and calm (unless you don't want to see them... so if you are hiking where there might be mountain lions or rattlesnakes, maybe better to make a lot of noise so they leave you alone). Sometimes the animals are all around you and you won't even know it if you are making noise or talking or walking too quickly.
For example, on another jungle walk in Borneo (in 2000), I was walking with a guide and there was another group of people who were chatting and laughing as they were making their way through the trees. So I asked my guide to hold back and let the others get some distance from us. After only a few minutes when we started walking very quietly again, we heard a commotion in the trees above us. Turns out we were surrounded by a troup of endangered proboscis monkeys (this photo is from 2018 again, due to the better camera than what I had in 2000).