In general, "hunting" implies a high chance of failing to find, catch, or kill whatever/whomever is being hunted.
"Hunt down" implies persevering until whatever/whomever is being hunted is "caught" or "dead" (either literally or figuratively). Often, this involves techniques to "corner" the target(s). A target that has been "backed into a corner" can only escape by going toward the hunter(s).
"Hunt down" is also more likely to imply that particular targets have been identified. For example, "hunt down men" might mean that:
- "she" has particular men in mind, or
- upon arriving at the party, she notices particular men, and will later on "hunt them down", or
- as men arrive at the party, she notices some of them, and will later on "hunt them down".