The Jensen omni is crap, I also have one and hate, hate, hate it.
For me, the problem with the conventional "Bat-Wing", is the ceiling is too high for even me (6ft plus) to reach it.
Wingard makes (used to?) a motorized Batwing. That deploys and turns with a motor. I thought (hard) about that, but ended up moving on.
Testing done by people in this forum suggests to me that the Jack Omni would work (at best) marginally better then the Jensen.
Both suffer from the same problems, size and height. No antenna mounted low to the roof is going to work as well as an antenna with more height.
Also size matters, small antennas don't work as well as large antennas.
I ended up with an Omni sold by Radio Shack. This is much larger then the Jensen or Jack. But I mount it on a separate push-up pole on the rear of the trailer. Obviously not something I would do for an over-night stop.
There is a "old tech" Antenna that should work well, it's two crossed elements that fold up and go into a tube. This design has been around for years, but I believe it is still sold