it seems unreasonable to assume that a highly advanced civilization intent in colonization would invest the economic resources and risk the political or social resources to do so very distant to their own world when much closer, viable options are a possibility.
However, after those first nearby colonies have developed, some of their population will have an economic, political, religious/etc interest in setting up their own colonies slightly further out. And some of those colonies will spawn others yet further out...
Exactly as humans spread around the world by foot and canoe.
And with FTL it would only take a few tens of thousands of years to expand through the entire galaxy, even if colonies developed fairly slowly (by human standards.) Without FTL it would only take a few tens of millions of years.
[And realistically it would happen faster. Colonies which develop quickest are those more culturally likely to seed further colonies. And the fastest developing of those would be the first to seed the next round. The process would be self reinforcing. The culture of colonising would be amplified each round.]