I would run the coax under the carpet following the wiring channels along the doors and up to the rear winow. We did a lot of police cars (with antennas on the roof) and usually fished them down the rear quarter panels.
I assume you are going to use an "on glass antenna" we used to come out of the box tuck under the head liner to the quarter panel and down under the carpet, crossing over the passenger side under the carpet into the console.
"On glass" antennas usually dont have the coax connector installed, and yes does your hang up have antenna jack