I think it would be great if then next Stargate series would be about disclosure to the public! They could call it "Stargate Worlds" or "Stargate Alliances" (names have been partially used before but I think the titles would match the theme).
The first season could be focused on the gradual movement towards disclosure and showing the public's reaction as the president makes the announcement on TV. Then they can add all sorts of elements including the NID, spaceships like the Prometheus and Daedalus, as well as travel to a variety of worlds. Perhaps Earth acquires a second stargate which is operated in a public area some place.
I think there would be a ton of potential for stories and so forth.
The most likely next stargate series will be called just "Stargate" and is a reboot of SG1 and the movie, with no SGA or SGU IMO.
Pure speculation. You're assuming there will be another series. What if it's just over?