tl;dr: About thousand years of guerrilla hideouts combined with some stranger projects (Thukker) and a hundred-year rebuilding spree (mostly diverted to build titans, though).
My personal theory is that both (Rebel) Matari and Amarr colonized worlds. The Minmatar were close to interstellar travel by the old version anyway, except by doing them in a "special" way. Even if they were completely spanked by the Amarr in the initial raids, I assume the Amarr did notice on early invasion that they had bitten more than they could chew - but, in true Amarr fashion, aborting the invasion was not an option. So they did what they could with their under equipped fleet, possibly designed to take out an industrial civilization, not an interplanetary one, captured a lot of slaves, ACN spins it as a fine victory, and they'll come back later.
However, it takes but a few smart Sebiestor, one downed Amarr slaver ship and some duct tape to jury-rig that Amarr drive into the remains of the colony ship, and off you go. By the time the Amarr came back, the Minmatar had prepared to go guerrilla on their current planets as well as possibly send a colony or two on other planets. Enter a thousand years of cat-and-mouse. The Minmatar, if being careful, might even get a head start of a century or so by picking "less suitable" worlds for colonization (avoiding detection), where the Amarr would spread slaves, possibly from the closest source, to the better suited ones.
And anyway, if some Imperial Exploration team claimed they had seen a non-Amarr starship, who would believe them? Everyone knew that the God's Chosen were the only civilized spacefaring nation in the universe. Or well, some monkeys from "Matar" had been able to use a giant slingshot and some chemical rockets to propel them from rock to rock, but such of a primitive technology could not take them to the stars, specially now when the ancient relics they had used to move between their home system and two others were securely in Amarr hands. Sure, the savages had gotten lucky and downed a few Imperial ships on some raids, but those ships were dead. the Emperor's engineers could likely repair those if they were moved to a well-equipped wharf. It is completely impossible that the primitives could repair the technology. In the matter of fact, a nation that barely mastered nuclear power and other such simple technology basically one step up from making fire would only burn their fingers trying to touch holy faster-than-light (or fast-sublight) technology.
And when someone really finds out that hey, these guys have escaped. The Imperial Navy (designed to look good, move to planetary orbits, bombard the place, and launch dropships) gets to play cat-and-mouse with the "Free Minmatar" forces. Possibly still winning any pitched battles with those siege tachyon beams, but then again, the Minmatar don't have anything yet that could find a "pitched battle" against the Golden Fleets, so they're mostly evading and making quick raids. The Amarr of course start to react, creating such innovative and agile designs such as the Augoror and the Maller.
But not fast enough - Minmatar have now experience on how to colonize a planet, and where the Amarr forces explore new systems likely still at the old, rather leisurely place (they're the only spacefaring civilization, remember, no need to hurry), the Minmatar are driven. They need to escape, or they're being crushed. One group takes this to the extreme, never settling properly and escaping far further than the other Minmatar ever, to the Great Wildlands. Some other groups find space closer, or are given some by the Amarr (Nefantar tribe home constellation, anyone?). Slowly but surely, the Amarr are catching up, though, and the hope seems lost. And then someone sees a green starship, has a chat with its captain over a few glasses of wine and the rest is history. The Federation is at war with the Caldari rebels, and they hear of a great Empire that is conquering all. Better help the small guy stall the Empire and try to finish the business with the Caldari. Of course, the Amarr have by now figured out what the Minmatar were doing, but they - not even after they have made contact with the Gallente themselves - likely cannot comprehend that the Minmatar now have access to Gallente technology, as well. Might take a moment to figure out the Gallente, too. What do you mean the Senate doesn't commit suicide after a new President is elected?
When the Empire retreats after the Rebellion, the Minmatar "start rebuilding" with the help of the Federation, but as evident from relatively recent information, they're partially still in the guerrilla mode, and divert a lot of resources to the Thukkers, who build them titans.