From SLC, one could take I-15 to I-40 then cross over to UT24 to Hanksville, but it is definitely much longer of a trip. US-6 from Spanish Fork to I-40 is four lane for a good portion of the drive to Soldier Summit, and is 65mph the whole way except near Price.
I-15/I-40/UT-24 : approximately 369 miles
I-15/US-6/I-40/UT-24 : approximately 231 miles
There are plenty of other variations to these too if one wanted to mix it up even more.
What are current conditions in the Henry Mountains? Is there snow and is it deep enough to require snow shoes? Any advice appreciated.
I really can't give you an update on the snow in that area but I can assume that the area did get some new snow after the storms that have passed through the state in the past week. The only person I know that has hiked Mt. Ellen was Erin (Wired) back on April 7th. Here is a link to her fascinating trip report: http://www.walkingwithwired.com/2015/04/day-16-mt-ellen-in-wind.html
You could also check with the BLM office located in Hanksville.