It takes about 30 seconds to receive the ephemeris data. For modern all in view receivers that is the limiting step. 30 seconds or so after power up and you should get data.
When I bought a Big Red Bee I looked at the heat shrink meant to keep the battery in place but also covering the patch antenna and realized that would cause trouble. Without the heat shrink it had a position 31 seconds after a cold start. It took 45 to 50 seconds with the heat shrink. The heat shrink changed the center frequency of the patch enough to degrade performance. The heat shrink wasn't all bad news since the RF inputs on GPS engines can be extremely static sensitive.
Another thing to worry about with GPS is that the national ranges will occasionally test weapon systems in a GPS jamming environment. The jammers of course cover a lot more space than that. A NOTAM is typically issued when this happens.
Could cutting a hole above where the gps pin is bring back that performance? I heat shrink and epoxy my gps modules and other components.