I don't know if the correction GPS will see the same error and thus give the correct correction factor or not.
Atmospheric conditions cause errors that are in some cases hard to predict.
If you have a GPS receiver at a known nearby location, you can subtract the difference from what you receive and what you would expect to receive at your actual location, and then add that in to your GPS receiver at the unknown location.
It needs to be nearby because you want the atmospheric conditions between your fixed receiver and your moving receiver to be as similar as possible.
See https://en.wikipedia.org/wiki/Differential_GPS for more information
Also https://en.wikipedia.org/wiki/Real_Time_Kinematic is what's used to get ~1cm accuracy, but RTK would not be useful because while RTK increases precision of the measurements it does not (by itself) eliminate errors introduced by the atmosphere, which are already larger than what can be obtained without taking carrier phase into account.
In the relatively near future, GPS satellites will broadcast multiple frequencies for civilian use, once commercial receivers catch up everyone should be able to reduce the error due to atmospheric conditions
I can't imagine the signals play that well in a heavily-jammed environment.
High probability rough GPS fix + radar / optical terrain comparison & dead reckoning for terminal guidance seems more likely.
I know they turned off Selective Availability ( https://en.m.wikipedia.org/wiki/Error_analysis_for_the_Globa... ) in 2000, so not sure if it's been true since then.
From what I was reading, it seems like semi-codeless approaches to the P(y) (aka military) signal offer approximately similar accuracy to actually being in possession of the key.
Furthermore, the P(y) signal uses the same L1 and L2 frequencies as civilian devices. Which makes sense when you're designing a system with 1970s broadcast technology.
From Wikipedia, it seems like the soon-to-be-launched Block IIIA satellites are the first that will incorporate the ability to spot-beam at higher power ( https://en.m.wikipedia.org/wiki/GPS_Block_IIIA#New_navigatio... ).