Hopefully they could pull a story out of their asses that is not inevitably affected by ME3's ending.
Wall of text incoming
I had a thought, maybe prior to the reapers nicking it, a Colonial Expedition is sent through the citadel relay, but not to darkspace ( somehow they managed to re-align it to set a new destination or something) but to another galaxy, say Andromeda, as a last resort, so that if shepard fails, civilization might survive, and the expedition might return to the milky way to rebuild after the reapers are gone or defeated. Every council race sends 25-50 of their brightest and strongest members. The protagonist would be a member of this expedition, but set about a century after the expedition arrived. They have built and populated many worlds, and are growing, they have met species local to the galaxy and joined the community.
The new trilogy would begin with a message received at "Ground-Zero" the expedition's capital and first landing site. The message was sent from the milky way, giving them the "all-clear". Due to the vast distances, calculations would indicate that the message was sent decades ago. A team who's ancestors were the leaders of the original expedition would be formed with the task of facilitating contact with the denizens of the milky way. This team, known as "The Descendants" and their leader (the protagonist) are sent back to the milky way. They are able to do so due to have acuired technologies from the locals, who have advanced far beyond anything we could imagine, due to not being subject to reaper culling.
Upon reaching the milky way, they are at first welcomed by the locals. Their advanced technology gives them total authority, for society has collapsed and they are the only people everyone is willing to follow, after spending most of the game cleaning up the galaxy, since the destruction of the relay system thoroughly ruined everyone's day, plans to open up the two galaxies are laid.
The second game is set one or two decades after, now travel between the two galaxies is common place, and the milky way is better off, but Andromeda is still superior. Andromeda's natural resources are dwindling, however technology sustained the population, no major advancements were made in millennia, the galaxy was stagnating. The Andromedans decided to exploit the milky-way, trading scraps of technology for entire star clusters. A clear societal divide is formed, with the lower class working in the milky-way and the upper class lazing around in Andromeda. Eventually, the original Andromedans deport all people whose origins lie in the milky-way, except for descendants of the expedition. Exploitation of the milky-way continues.
In the third game, those council-races still in Andromeda have completely alienated themselves from their origins, and downright deny their roots. Very few still embrace the milky-way as their true home, and even few speak out against it's exploitation. With over 50% of the milky-way strip-mined,
The lower-working class is getting more and more reasons to revolt, but they cannot due to Andromeda's superior tech. A direct descendant of the protagonists from the previous two games re-creates "The Descendants" as a terrorist group fighting for the freedom of Milky-way. Their actions, such as arming the workers and supplying them, leads to an all-out war between the two galaxies, and the outcome depends on your choices from the three games.
This is a very rough idea, and bleeds from many, many plot-holes, but i think it's doable, and would make for an interesting story