Also, side note, if you already signed paperwork with the new employer, press as hard as you think you can get away with, as soon and frequently as you can.

Squeaky wheel gets the grease. Sometimes, there's nothing they can do (hiring freeze from the C-level); but other times, they need to turn 10 hires into 3. Those 3 typically end up being the ones who push for it.

Would definitely push on both fronts. Good luck!