I think what you mean when you say headphone jack is headphone plug. The jack is the receptacle; the plug goes into it. And by Roger base unit, assumedly you mean docking station.
I have a round Roger Select iN microphone, which is the model previous to the current (and rectangular) Roger On iN. (They’re gonna drive us crazy with these names.) It came with several cables. One has the small USB-C male connector on one end, which plugs into the microphone. The other end is a 3.5mm stereo type (three contacts, i.e. two conductor shielded) male micro phone plug which plugs into many things including airplane audio jacks. If its one foot length is sufficient (you can tuck the microphone into the seatback magazine pocket by the tray, from which it will transmit to your hearing aids), you’re done – unless you want to power the microphone externally rather than relying on its internal battery. In that case, you bring along the docking station. Insert the microphone into its docking station. At rear, plug in a 3.5mm male to male jumper cable of suitable length, which you bought unless one of those came with your microphone. If necessary you can lengthen that cable with another, this one a 3.5mm male to 3.5mm female extension cord. That connects the airplane audio jack to the audio jack on the back of the docking station. The Roger microphone also came with a power cable to use with it or with its docking station. In this case, since the microphone will be sitting in its docking station, the USB-C end of the power cable now plugs in at the back of the docking station. The other end of that cable is a standard, larger USB connector. Plug that into a powered USB jack to get the needed 5V power, if one is provided on the plane. Otherwise plug it into a power adapter which you brought for whatever type AC power connector is available, to convert the AC power socket voltage from 110 or 220 down to 5V as required by the docking station. You can tuck the docking station and unneeded length of cables into the magazine pocket, or leave it on the open tray table. Just don’t let it hang in the air to avoid stress on the cable connectors or worse, having the microphone fall out of the docking station and end up lost under a seat. Now you don’t have to worry about the Roger microphone running out of juice on a long flight.