Look into a progressive strength program like Starting Strength or Practical Programming, Grey Skull Linear Progression, etc, then adapt it to focus on your bench and not stress the other lifts as heavily if you don't want.
The machine you're describing sounds a lot like the "Universal" machine we had at my high school weight room way back. It looked a lot like this one, note the bench station on the right of the picture:
Universal Machine (WFS)
This set up had the bar at deficit below your chest line unless you were really skinny.
Since that machine starts you out low, work on your mobility. If you're struggling to get under it, you won't have any power to start the weight moving.
Most progressive programming won't have any accessory work that would let you do this, but if you want to work on getting that low end push developed to get under that machine, deficit DB presses that only go up 1/3-1/2 are good. Negatives build strength quickly also, but they tend to need a bit more recovery time. The best way to increase maximal strength is to spend time moving heavy weights.
Ultimately, like I said - get on a progressive program and stick with it and FOLLOW IT TO THE LETTER.