Water – well pump will not fill pressure tank

That sounds like an overly complicated setup with far more pumps than it should need. Or else you said pump when you meant tank in two places, actually, re-reading it.

In short, you can have as many pressure tanks as you like. Locating them closer to the point of use (the house) would be helpful, to a limited extent. That limited extent is the "effective volume" of the pressure tank at the house - while it's got pressure, you get better pressure at the house. As soon as it runs out of water and you become dependent on the pump, the pressure loss of the 300 feet of 1" PVC to the house comes into play, until you slow down your water use to the point where the pressure tank at the house fills up again.

If you do not change the setpoint pressure, the peak water pressure in the house will not be affected at all. Once the pressure tank at the house is depleted, the pressure will be the same as it is now. If you have a pressure gauge at the house now (or can add one) it might be useful to know the pressure at the house when "loads of interest" (showers, say) are drawing water.

Most pump controls can be altered to have a higher setpoint pressure, and most well pumps will provide that, within reason. General recommendations are to stay below 80 PSI - but if you are now running your pump on at 20 off at 40 (a typical default) you could turn it up to on at 30 off at 50, or on at 40 off at 60 - you will need to adjust your pressure tank (and any you may choose to add) for the higher pressure, particularly if they are bladder-type tanks - with the system drained, a bladder tank should typically be 2 PSI less than the low water pressure setpoint (ie, 18, 28, or 38 for the three ranges I've just given) and you may need additional pressure tank volume (because the effective volume of water a pressure tank can hold goes down as the system pressure goes up.) So, you can probably get more pressure without another pump, and possibly without another tank, but we'd need more details of what your system is doing now to know that for certain.

Unless your water use is extreme, 300 feet of 1" PVC pipe should not have a lot of effect on the pressure - at 5 gallons per minute, about 2.2 PSI - at 10GPM, 8.2PSI, 15 GPM, 17.4 psi

Allowing for the possibility that it could be otherwise as diagnosis via internet is imperfect....

I see both a pump problem and a check valve problem here. While they may be located in the same place (if your only check valve is on the pump - which is the way mine is set up) they are not the same thing...though I also see another possibility that would do both.

• Pump problem - pressure not getting above 27 PSI, or 15 with a faucet open.
• Check valve problem - when you shut off the pump the pressure drops to zero - unless you are using water, the pressure should stay at 27 if it was pumped up to 27. When you only had 10 PSI in the tank, there should have been some water storage - when raised to 38, no water could possibly make it into the tank, since that's designed for 40/60 PSI operation and won't take any water until the the water is over 38 PSI.

The third possibility - you have a leak in the pipe above the pump and check valve - so the pressure is limited, and the water drains away from the leak without the check valve being to blame. If you can go and listen at the top of the well while someone else turns the pump on, you might actually hear this leak, if its in the well above waterline. Or you might have a recently soggy spot in the yard if it's after the pipe leaves the well.

While I don't think it's to blame in this problem, I'd also suggest rechecking your tank pre-charge to see if it has stayed at 38 PSI or started to fall again. Often when the precharge becomes low it's not simply a matter of needing to add air - it's a failure in the bladder or diaphragm.