I had a 2010 JKUR stock and did not enjoy towing with it. I live in the mountains of eastern AZ. Driving it empty in the mountains annoyed me due to not necessarily the lack of power but the spacing of rpm between 4th and 3rd. If I put on cruise at 65 mph it would shift from out of OD (4th) at the slightest incline, but not after dropping to about 59 mph, then sometimes drop to 2nd screaming to get back to 65. This would yield about 16 mpg. If I anticipated the hills and take from 4th to 3rd before the incline I would get about 17 to 18mpg. Mind you this was empty.
When I hooked up my sierra 4x4 trailer appx 1800-2000 lbs and loaded the inside of the jeep with a fridge, two dogs and a baby, 4th gear basically disappeared except on level ground at 65 to 70 mph ( minus any headwind). If the horizon had the illusion of an incline it would shift into 3 rd. this would yield 14 and sometimes 13 mpg.
I really don't fault the 3.8l at all, it's a great engine for durability. It's the gearing which is not optimal. If it had a 6 speed it would be ok. You can easily tell that the power curve just doesnt match for the speeds and mountains of AZ.
Coming out of roosevelt heading east there is a long uphill grade and I could do it loaded with trailer at 55mph in 2nd gear. Sometimes the AC would kick out depending if it was the dead of summer at about 7 mpg. While this is tolerable (Barely) it is very annoying. very inefficient system. I could have 5th wheel diesel rigs with a second trailer pulling a boat or utv pass me going 60 mph and most likely getting double mpg. It always felt like that I was driving at 100% power just to go the speed limit in the hills. In this day and age 55 mph and 65 mph should not be a problem when towing at 60% of the vehicles weight rating.
Since I have the off road trailer and we love traveling in it, I decided to get a 2013 JKUR with an auto. If you look at the torque curves between the 3.6 and 3.8 the 3.6 is flatter through the rpm range. Combine that with slightly different gear ratios and one extra gear it is noticeably better.
Just my .02 worth. I really enjoyed the 2010 but the 2013 is much nicer.