I'm gunna go out on a limb here and guess that there is no easy fix or secret trick that will resolve your issue; you'll need to do a little troubleshooting.
- The first thing is to try a different USB stick. Beg, borrow or steal a different brand/model USB stick. Put a podcast on there, use it for a few days, see if that fixes it. Alternatively, you may consider re-formatting your USB stick and trying again.
- Perhaps your particular USB stick has an internal issue with the connector which causes is to lose connectivity when not in use.
- Your USB cable in the Fit may need to be replaced (under warranty).
I'm not sure exactly how the head unit "remembers" where you left off on your USB drive, but it may do this by providing a consistent unit load (100mA, 5V when powered up, less when not up) indefinitely (even with the engine off). If this is true, the source of your issue could be your head unit or your flash drive. Between these two, they are losing connectivity.
Again, try a different flash drive first and see if this resolves your issue, then move on to the reality that your USB cable (at best) or your head unit (at worst) may be malfunctioning.