I’m okay with saying that the First Order has risen from a single base, since that is all that we see in the film. Granted they have more, but they’re not important for this particular story. I also want to include the base because the Starkiller needs some reference in the crawl so that it doesn’t come out of nowhere later on.

I do get that it is weird that the First Order would only start conquering things after Luke went away and from the one base, so I can change the wording to suggest that they are merely announcing their supremacy in this area, which leaves the actual conquering of the systems beyond the Republic to the imagination. For example, one can imagine that they have been de-facto in control of these territories for decades, only deciding to formally announce their control after Luke disappeared.

EPISODE VII
THE FORCE AWAKENS

It is a time of despair.
Luke Skywalker, striving
to reforge the legendary
Jedi Order, has vanished.

Emboldened by the Jedi’s
absence, soldiers of the
evil FIRST ORDER have
risen from the shadows
of an Imperial fortress
and claimed supremacy
over every world beyond
the fragile New Republic.

Desperate to save these
beleaguered planets, an
armed RESISTANCE loyal
to the New Republic has
sent its pilots in search of
the last Jedi, seeking to
restore the light of hope
to the darkening stars….