DO NOT UP YOUR OFFER UNTIL THEY ASK FOR MORE. It doesn't help; they still reject it, and you restart the time cycle in which they respond to your offer by asking for more or whatever.
Here's how the process goes for anyone who is not labeled, "looking to sign:"
1) You make the offer.
2) They respond with some sort of vague message.
3) You wait 3-7 real days until they get back to you. This is very important. Do nothing until you get their second email.
4) They will either respond by:
a) Signing, or
b) asking for more money, or
c) rejecting you outright.
The probability of each of these is determined by their "signability" in your background info on your draft list. We'll get to that later.
5) If they sign, cool.
If they ask for more money, if you offer that more money they will 100% sign.
If they reject you outright, DO NOT DELETE THE REJECTION EMAIL. There is a link in your inbox next to the rejection email; click it, and you can offer them the same money again, which will renew the cycle.
Signability:
A) If their initial signability is, "looking to be drafted in round..." it is my experience that after the initial vague email, they will either just sign for your initial offer, or ask for more money.
B) If their initial signability is, "may sign..." or "undecided..." they almost always will ask for more money; they will rarely sign for your initial offer, and they will slightly less rarely reject you outright.
C) If their initial signability is, "probably won't sign..." then there is about a 95% chance that they will reject you outright.