The snow was retreating quickly then the upper elevations got 12-18” just before Memorial Day, but that should melt quickly. Most the snow below 9k’ has melted. I’m guessing anything with a south exposure will be clear below 10k’ by mid June, patchy snow 10-11k’ with larger snow fields above that.
Here’s a nice lake a few minutes outside Salt Lake valley at 9k’. Check the photos for snow the past week:
https://www.alltrails.com/trail/us/utah/lake-blanche-trail?u=i
The Mirror Lake Highway through the Unitas is open and there are several options there that will have fewer people and better fishing. The Naturalist Basin is nice. You’ll probably have patchy snow at higher elevations but snow shouldn’t be an issue other than the high passes, which this avoids. There was a fire in the area a year or two ago but I haven’t been there since so not sure of the scale of deadfall etc (all-trails trip reports have info):
https://www.alltrails.com/trail/us/utah/naturalist-basin?u=i
Just south of there is Granddaddy Basin, which most access from a different trailhead than Naturalist Basin, but can be reached from either. There are also a number of shorter routes on the west side of Mirror Lake highway from trailheads at Washington or Trial Lake to lakes like Marjorie or Duck that might work: https://www.alltrails.com/trail/us/utah/duck-lake-trail?u=i
Hope that’s helpful.