How big a tank? Neons can be a snack to angelfish as they get larger.
You've got the room for some more fish, I take it you are looking for some sort of centerpiece? My tank is only a 20 gallon so i look more for schooling fish, but your could use a pair of honey gouramis. Angels would work and fit in with the Amazon them you have going but they are also cichlids and get temperamental as they grow.
There are some other dwarf cichlids that would work, apistos and other rams, provided they all have their space. A small pleco is possible too, a bristlenose pair or maybe a clown pleco..something that stays under 5 inches.
What temp are you trying to keep the tank at? 79F is too cold for GBRs (they're very disease prone in cooler waters sub 82F and have poor long-term survival rates when kept that cold), but it's also above the comfort range of Neons (68-78F). Rummynose tetras also prefer the 73-79F range, so if you make the Rams happy then the tetras will all be unhappy. Keeping it warm enough for the GBRs will shorten the lifespan of the tetras since the warmer water speeds their metabolism.
Rams and Angels get along fine, but depending on the Angel's personality it may try to eat your neons and rummys as it matures. Some accept smaller tankmates that they've grown up with, but not all.
Ditto to elephantnose, give the Rams what they need or the chances aren't good for them. The tetras can handle the higher temps better than the rams can handle the cold, but just recognize that it knocks a few years off their lifespans. They really shouldn't be kept together long term.
A few years? o.o how long are their average lifespans?
Well cared for Neon Tetras can last up to 10 years, and 5-6 for rummies.