The fed does indeed sometime draw in money and park it (to be used the next time they need to increase the money supply, in a sense)
It's also the case that money velocity can always slow to a crawl. If it reaches a point where people figure they are more likely to keep value by just holding the money instead of investing it into something due to falling asset values, that can overcome even the desire to beat inflation. It's all very situational, but sometimes money doesn't get invested, or at least the rate at which it changes hands slows down due to a lack of good investments in the system.
It's also the case that money velocity can always slow to a crawl. If it reaches a point where people figure they are more likely to keep value by just holding the money instead of investing it into something due to falling asset values, that can overcome even the desire to beat inflation. It's all very situational, but sometimes money doesn't get invested, or at least the rate at which it changes hands slows down due to a lack of good investments in the system.