Since this is a plot point that has been constantly debated throughout the entire run of the Stargate franchise, here's something I would change:
Make the Stargate public knowledge from the very beginning, and actually allow the world to react to the news instead of keeping it a secret the whole time.
Pretty much it at this point.
They had the main show that started it all with Special Force teams exploring the galaxy.
They had the spinoff that moved to a different galaxy with the spin that they were cut off from their main support (at least for a while).
Then they tried their hands on their BSG version (which i personally believe was a huge mistake).
They pretty much got everything covered that they can get out of the premise before they have to repeat themselves and have the villain of the season style show ala Buffy and Angel.
The program still being kept a secret is what bugged me after certain points.. they had big ass spaceships in orbit, huge space battles and at one point disappearing skyscrapers from the middle of a city and the world is still believing everything's normal?
Personally i think the premise of the world getting to know all of it could be quite interesting but the way i envision it it may be a bit too serious/political/social commentary in tone to fit in with the rest of the show which is more light weight.
However with the right writing staff and show runner it could be an interesting look into how the world could adapt when thrust into this.. the initial chaos, the power plays and on top of that some main villain for the action part.
However i believe Stargate is dead for the time being as much as Star Trek is dead for TV. It may take a few years more for it to be given any shot at all.