So if you're leaving Jackson Friday morning, and your goal is to stay in West Yellowstone Friday night, and be back in Jackson Saturday night...if I'm reading that right, that basically give you two FULL days in the park if you want. Your park pass is good for a week and your fishing license will be good for 3 days minimum, so you might as well get in a couple days worth if you've got the time and ability!
You're only about 1.5 hours (maybe less depends on how fast you drive) from Jackson to the South Entrance of the Park. So depending on how early you want to wake up and get on the road, but for sure you could be there by lunch without much effort. There isn't a real easy way to just go all around the park, you'll be back tracking pretty much no matter how you try and do it. Most of my favorite places to fish in the Park are in the north and north east areas. To qualify that, I have spent a lot more time over there, so the other places could be just as good if not better.
Here is what I'd do if I was in your situation (assuming your traveling partner is game for some fishing!) Enter from the south and focus on the Firehole and/or Madison that day. You could see some of the sights like Old Faithful as well without too much of a detour. Check with some local shops on hot spots and flies for those a couple days before you head out. Watch for the temps on the Firehole, they shut it down when the temps get too warm. These are easily accessible from each other and could give you an hour or two at each place while seeing some of the sights as well.
For your second day, wake up EARLY and head to the NE portion of the park. You'll see TONS of wildlife on that trek. On your way stop in Mammoth for fun and also check out the Gardner River in the area. Funny story last time I fished it, my buddy caught a cutt, bow, brown, and brookie all on that river in 1 day. He had a 20+ fish day easily. I got skunked. I can't explain it. Just one of those days. It's a lot of pocket water, but fun to fish. Then head to the NE portion of the park (my favorite) and the options are many. Wildlife in this area is insane. Bears, wolves, bison, elk a plenty. I'd stop in at the Lamar River and toss a line, and proceed up the road to Soda Butte. Soda Butte won't have huge fish, but is one of my favorites there that are along the 'beaten path.' Right along the road most of the way, and fun fishing. I would suggest Slough, but it's more of a day thing, not an hour or two thing. Then you can swing passed Yellowstone Lake on your way out the south entrance and back to Jackson that day. It will be a full day, but most days in the Park are.
I think the most effective way to fish these rivers in Yellowstone is figuring out what's going on with the hatch and matching that stage exactly. Before you go off and say, "No crap genius!" Most anglers don't do this on a regular basis. They just see adults and throw adults, or if not, throw general nymph rigs. Yellowstone isn't like most our rivers around here. Often, matching the stage of the hatch is necessary to even catch a fish. There should also be terrestrials, and that is a fun way to fish.
It really is one of my favorite places to visit and fish. So many options. Get that book I linked you to if you can and it will give you more information than you could probably do in a lifetime up there. But I'm going to have a good time trying!!! [
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