Goosegg,
If you are a senior, you want to be signed at the first possible day after the draft, unless you were picked in the first 8-10 rounds and are negotiating for what might be a significant bonus. However, senior signs have little leverage and with the new slotting system, most receive a take it or leave it from the drafting team. Most seniors want to be signed before about June 20 or so, as I will explain.
The reasons most seniors want to be signed as quickly as feasible are at least two: the first is they don't want the team to fill up their draft allocation and have adequate players signed at the position of the drafted player to fill the available MILB slots. A senior who is still waiting as of July 15, will be in jeopardy of getting any offer. A senior needs to remember that the short season rosters are compromised of not just drafted players but also those assigned to extended Spring Training and a good number of Latin players and others who are not part of the draft.
The other reason is the short season leagues start around June 20. A senior sign is going to want to get an early assignment and be playing and hopefully to the highest level possible in the NY/PENN or Northwest League. While there are no guarantees, the early senior sign will get a roster slot in most likelihood in one of the short season leagues. A senior who might sign in mid-July or August is going to miss nearly an entire season, which will probably heavily impact any chance of them getting a contract the following year.
Hope this helps.