I've recently worked out the biological father of a friend and a couple of odd matches within DHs and my family which are clearly the result of sex outside marriage.
A genealogist may or may not be able to help you at this stage.
If your closest match is a second cousin once removed you possibly will end up with a situation where you can't establish which sibling it is even with the additional records, you don't have.
If you have several possible brothers, it then comes down to where they were at the time you were conceived. Some of this may be in records but not necessarily all. The experience of the genealogist may come in here - many will just use records available online. Some may look into locally held archives not available online.
It's often fairly easy to track someone down cross referencing the gro birth indexes and the electoral role but it does depend on where in the country they live.
The genealogists on TV shows go further - they often track down the family, make contact and ask questions. There's also a problem here though.
For every case you see on TV they will have researched another whether approaches to family are either rejected or find out nothing useful.
This is where you need a closer match than 2nd cousin once removed in your matches.
If you can find someone you think is closely related and get them to do another DNA test to give you another reference point.
It's worth keeping in mind that when they track down relatives on TV shows, they often have cases on file for some years, whilst they wait for a good match to pop up before they pursue further with the research.
That's why I say it's not just about hiring a genealogist: it's a combination of a good genealogist, a bit of luck and/or cooperation from the family and patience all coming together.
Not everyone in your situation will get that unfortunately.
I wish you luck in finding out. X