The pump rod has a pretty good amount of adjustment available. To be clear, the pump rod adjustment is done on the end where the pump rod screws into the plunger guide https://www.facebook.com/photo.php?fbid=309886555726284&set=a.309884835726456.70061.207559939292280&type=3&theater
Loosen the nut on the pump rod and then rotate the pump rod counter clockwise to lengthen the pump assembly. Get the pump to the point where the pump just bottoms out on the valve chamber. Then, set over toggle: by loosen nut again and rotate the pump rod about 1/4 turn additional counter clockwise.
If there really is no adjustment left, then there has been a tremendous amount of wear in the pump linkages. I would inspect the entire pump assembly for excess wear.
Note: I would never assume that a check valve is working properly, just because it has been recently serviced. Things can go wrong very fast. A loose bit of metal can get stuck on the valve face. etc.
From your description, it sounds like the pump arm is being pushed down by air pressure not just falling down. That's indication of a leaky check valve.