It might be fruitful to search either by index reference, or by first names and district only, to find any other surnames mentioned.
This could throw up possible marriages, or at least give a clue to what was happening at the time.
In my ONS, I have found children initially registered with the same surname as the mother, then re-referenced later, with a different surname, as well as the original, in the later index. One child had two such father figures in their life, and a third re-registration following the mother's marriage.
In earlier indexes, there are often cross-references inserted. One of my relatives was born in 1914 before her parents married. There is another entry around the time she would be starting work, this time with her father's surname. This index refers back to the original one, where there is a handwritten comment.
Nobody said this was going to be easy!