I haven't been to 8-mile canyon. It's well over 100 miles from me. I went on the ATVIdaho link that was previously posted and the trails listed there that I've been on are Garn Mountain, Kelly Canyon, Relay Ridge (actually we have a different route to Relay Ridge that comes in from the east that would not be rated beginner!), and Felt (which we call Jackpine Loop). There's also a dozen or so other trails in Teton Valley not listed on that website.

Did you notice that the website indicated that 4X4 was needed on the 8-mile canyon ride? I saw in your OP that you were looking for something that didn't require 4X4. The old railroad bed that 24mileboy mentioned is the one that comes all the way down here and crosses Bitch Creek at the trestle shown on the "Felt" trail listed on the atvidaho link.

You may luck out with the snow situation by coming in mid-June since we haven't gotten a lot of snow this year. There definitely won't be any over by St. Anthony, but there could still be some in the Tetons, Big Holes (where Kelly Canyon and Relay Ridge trails are) and even Island Park. At the very least, the trails may have big puddles on them.