I think that the phrase 'big boys' should be defined more clearly. The largest companies in the industry generally take on complex, large contracts with a low net profit margin which SMEs wouldn't be equipped, or would even want, to service.
SME, though, covers a wide range of business sizes ranging in turnover from £3000 to £30m and there are certainly 'big boys' in that - these companies have mostly become larger through years of trading and I don't think that there's a realistic shortcut to that process.
My reading of your post, which may of course be incorrect, is based on the assumption that other cleaning companies view others as 'competition' on an individual basis and I struggle to accept that as being accurate. When I'm tendering for a contract, I know that I'm going to be bidding against 2-5 other companies and I really don't care who they are; I've never been particularly interested in the size of a company who I lose a contract to or gain a contract from and I never think about how my business success or failure affects other companies.
So my answer to your question is this:
I don't think that SMEs are 'on the floor'; we're sitting at a larger table with better food. Not so much with the caviar, certainly, but there is more meat on our chicken and our fruit is considerably smaller but has much, much more juice when you squeeze it.