Not quite, the flood originated from the precursors in a supercluster near the ridge of the galaxy, they were not detected going anywhere else. The were boarded on a ruined ship and infected working ships to spread. Then became the bastards we know today by a series of forerunner attempts to stop them.

The librarian found hundreds of thousands of ancient forerunner ships in the LMC. Millions of years ago the forerunners eradicated the precursors in that galaxy, but they never came home out of great shame. Surviving precursors made themselves into the powder stuff but over the ages it became corrupt (or perhaps it was always intended) and thus the flood were born. These are the flood that came to the milky way. It can be assumed the precursors dwelled in many other galaxies, but their existence in the milky way and the LMC is the only thing we know about. Forerunners themselves, after all, may be living in other galaxies as well. A small mention is made in the first foreunner book that forerunners could travel not only to any place they wished, but other dimensions as well. For all that, they were always less advanced than the precursors. So it is likely that precursors (flood) live in other galaxies.

Considering there prey had been destroyed by the firing of the halo array I doubt that they would have time to board a forerunner or prehuman space shuttle that would have been able to leave the galaxy

I don't recall that. Can you do a direct quote and citation?

They originated from the Large Magellanic Cloud. Curiously enough, it is not infested in the slightest, and any remaining traces of the Precursors there was destroyed when Omega Halo was fired at the LMC.