Grand Teton / Jackson / Yellowstone
June is my favorite month here in the Tetons. Great time to come visit; everything is green, the mountains are still snow capped, and the rivers are all running full.
Regarding Yellowstone NP, unfortunately there is no paddling allowed on rivers other than the Lewis channel connecting Lewis Lake & Shoshone Lake. This is a great section for canoes/kayaks but not a raft. Paddling across Lewis Lake is required to access the area. Otherwise you could plan a "protest paddle" on some of the other great stretches of rivers in the park...but you could be fined and banned for life.
Regarding Grand Teton NP, there is paddling allowed only on the Snake River. You mentioned a popular stretch, Pacific Creek to Deadman's. This is indeed a beautiful stretch of river with fantastic Teton views. This section is known to have braided channels and numerous snags/strainers in the form of massive pine trees. While not as challenging as the Deadman's to Moose stretch, it can be quite technical during June flows depending on the day and the location of the migrating strainers. It is probably class 2 water, but has challenges and dangers that require class 3 decision making skills in my opinion. Consider launching below the dam to extend the trip a little and paddle into Oxbow bend to take in the vista of Mt. Moran. Hitching for the shuttle is possible as long as you're not in a huge hurry.
The Snake River also has stretches in the Jackson area that are worth checking out including the Hoback River tributary. There are both scenic and whitewater opportunities. If you are paddle enthusiasts you could easily spend two days here on these runs.
Beware of required boat stickers. Yellowstone and Grand Teton both have their own required sticker and require the State invasive species sticker. Waters outside of the park require only the invasive species sticker. I highly recommend ordering these in advance if/where possible to avoid detours to ranger stations, lines, etc. Lastly, be sure to check river levels / conditions before you launch each day.