I have a new pump installed that is made for kicking on and off with one gallon tank (according to installer). Basically no storage tank needed. However, in pursuing an intermittent problem I had already installed a new big tank. My question is this: can I reduce the on/off limits to within 3 or 4 psi of each other? I am getting really low pressure just before it finally kicks on the pump. Seems like if the new pump was designed to work within a much tighter range, I'd should take advantage of that. I think the installer set the new pump to match the old style tank unnecessarily. He did say there is a computer board that would have to be added if I were to go to the "on demand" protocol the pump was designed to meet. Still...
Your water storage/pressure/booster tank could be low on pressure. I would check it out and make sure it is set for the proper psi as the pump. 50psi pump cut off out mean the tank should have 50psi in it when full. Ideally a tank should be a mix of about 50 air and 50 water. If you have trouble with this, your newer tank could have a bladder or balloon issue.
In process control terms, deadband is useful to prevent the pump from being turned on and off too frequently. The really tight deadband you propose would almost certainly make the motor wear out quickly.
A nicer solution would be to increase the overall tank pressure (say to 40 to 80 psi) and add a pressure regulator to the output, maybe set at 35 psi.
You appear from your description to have the option of (an option you have not bought, evidently) a "constant pressure variable speed" pump. That is the sort that uses a very small pressure tank - rather than turning off and on, it spins faster or slower to meet demand while there is demand - assuming you've bought the controls that allow it to do that.
Since you have not bought those controls, it's a single-speed pump operating at line frequency. As such it will have the normal issues with short-cyling and should be operating into a pressure tank that is set about 2 PSI lower than the "cut-in" pressure, with a 20 PSI differential. - so, for a 30-50 setting, the tank air is set at 28 (when empty of water.) If set 50-70, the tank air is set at 48. "Getting really low pressure just before the pump kicks in" is a sign that the tank air pressure may be set too high, as it will then run out of water.
Or you buy the variable speed controls, if constant pressure is something you are willing to pay whatever it costs for.