Sounds to me like you're just getting some amp hiss. They all do, but it shouldn't be loud enough to affect the recording. If it is there is probably something less than optimal in your gain staging. For example poor signal/noise output from your pickups, or the wrong gain/master settings on the amp.
You could just manually remove hiss from non played passages in your wave editor, if its not too loud to affect the recorded bass parts. EQing just to get rid of the hiss could have undesirable side effects on your bass sound - it's surprising how much activity there is up above 1Khz for example. The DI outs on many amps are not as good as dedicated DI boxes.
With a Bass Pod you're paying for a lot of different simulations, which is fine if you want them, but a straight DI box like the Sansamp Bass Driver (or something more upmarket) would be cheaper and might do all you need. I trust that you're using very gentle settings on your compressor - overdo it and you've got nowhere to go in mixing. The make up gain on the compressor will bring up the level of quiet sounds which would exacerbate the hiss problem.