Do the chuck and the arbor have the same taper? Chucks are nearly always Jacobs taper as is one end of the arbor. The long end of the arbor is usually Morse taper.Is the arbor taper and the quill taper the same? does the tang on the arbor fit in the slot in the quill? If yes to these, are there any burrs on either the chuck or arbor? Is there any cosmoline or other corrosion preventative on either one? If yes, remove and polish. Must be clean. If no, clean anyway. then, put the arbor in the quill and check runout. By snugging up the quill bore some runout can be removed but do not squeeze quill. IIt should still move freely. If still out, remove, turn 180dg and reinstall. Check runout again. If still out, thats not unusual for Jet. At this point, go to your local machinist supply and buy an arbor and chuck of good quality and have them mount it.
If the runout is acceptable, freeze the arbor overnight in the deepfreeze. Heat the chuck to 300 dg in the oven. Have the dogs in the chuck turned down flush or less. Have a brass hammer and a piece of 2x4 handy. slip the arbor into the chuck and give the tang end of the arbor a couple taps with the chuck set firmly on the board. If the tapers are true, this will seat it squarely. You have only a couple seconds to do this.
Good Luck. Personally, I'll never own another Jet tool