Hi
I’m not an expert, but here’s a few thoughts.
Have you done a DNA test as well as your daughter? From what you say, it seems maybe you haven’t, but this would be really helpful, for two reasons.
1. You are looking for your daughter’s paternal great grandfather, which must be your grandfather, if you are her biological father. Doing the test yourself gets you one generation closer, and that would be helpful. The fourth cousin matches to her should, in many cases, become 3rd cousin matches to you, which may make it clearer which matches to follow up. In my family history research, important relationships to me were not clear from the DNA, but became very clear when my aunt also tested.
2. It helps eliminate matches that she has that are from her mother’s side. I have never used phasing, but I understand it is a tool to identify the origin of parts of a person’s DNA when one or both of their parents have also tested.
If you are lucky, there will be a match that makes things very clear, but quite possibly your 3rd/4th cousin matches will not be clear without some paperwork. The only way forward I know is to try to find a paper connection between one of your matches and you/her. This will likely take some time, and lead up a few blind alleys before you find the one that works for you.
My suggestion would be to try to estimate which matches look most promising. The obvious starting point is people who have good family trees available, and who want to help (and so respond when you contact them). Then use a tool like “in common with” or the chromosome browser to examine some of your closest matches, to see if a group of matches match you/her on the same segment, thus meaning they may all be also related to each other. This isn’t certain without more information, but may help indicate a group that may have ancestry in common, and you may be able to use information from several trees to try to find the connection to your own.
Then I think there is no substitute for hard work, following your match’s trees backwards to a possible common ancestor and then forwards to see if you come up with someone in your own tree. I have tried this and found it a long and frustrating process, but it may be the only way, unless you get lucky or are willing to wait and hope that someone tests soon who gives you the breakthrough. In my case, it was a new tester rather than the hard work that gave me the breakthrough, but I’m still glad I tried tracing match’s trees rather than sit and wait.
Hope that helps.