In Yellowstone, Avalanche peak is always our warmup (we'll do it the same day as Washburn), and you can traverse over and get Hoyt along with it. Fun hike, not very crowded.
Electric Peak is a great journey with an intense find-your-own-way-up at the end, but its a pretty stiff day-hike (I think its 22 miles via Glenn Creek Trailhead and you have to effectively pack-in all of your water)
Bighorn peak is fantastic, it gets you on a bit of the Sky Rim trail, accessed from Dailey Creek Trailhead is a bit out of the way, but you'll feel like you own the mountain: we've only ever encountered one other party on this hike; but plenty of sheep!
Republic Pass (accessed from Cooke City) is quite a lovely hike in a not too traveled section
Baronette peak is an epic bushwhack...and a roller coaster...lots of up, lots of down, no real trail.
Trout Lake is a must--easy hike, plus the bonus of otter viewing if youre lucky! (downside: tourists and fisherman)
In the Tetons...
There is Garnet canyon (of course) which gets you to The Meadows 🙂
Buck Mountain is a fantastic adventure (and an excining climb at the end. Non-technical, but depending on the season, and axe and crampons (and knowing how to use them) buys a little confidence. The last 600' is quite exciting!
If you're going to take the Cascade Creek trail up to Lake solitude, it is of course lovely but a VERY busy trail.
Five star review for Lake of the Crags; it is is fantastic and very low traffic (even though it is basically next to inspiration point). We did this hike and encountered no other parties! I cannot speak highly enough of this hike!