If memory serves, Hobbit expansion throughout Middle-Earth happened like this: they appeared somewhere in the North of ME, East of the Misty Mountains. They then began to migrate down, through the Gap of Rohan and eventually up through the Shire. There were a few who dropped along the way;
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(The Hobbit)In fact there were probably a great many more outsiders (or something like that, meaning hobbits living outside the Shire) than Bilbo knew about...
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but most seem to have settled in the Shire or at least nearby. Gollum's Stoor clan never migrated, and stayed along the bank of Anduin. They were a rather more primitive people, obviously, for reasons stated, and had a society most unlike that of the Shire; matriarchial, without too many comforts, and interestingly giving birthday presents to the person whose birthday it was, not the other way around (Smeagol demanded the Ring of Deagol because it was his birthday). However they were definitely similar. They both lived in holes, were both quite long lived, and both hardy, resilent and down to Earth for a start. There were similarities in the TT between Gollum and Frodo, and between Bilbo and Gollum in the Hobbit, for a surety.
Gandalf said they were of 'hobbit-kind'. Well, we say we are Man-kind, and we are Men; I think that's what the wizard meant. 'Akin to the fathers of the fathers of the Stoors' could simply mean that it was generations back from the present day Stoors; in the days of the great-great-grandfathers of the Shire.
To conclude this somewhat scattered post. No, I do not think that the Shire hobbits had 'evolved' from Gollum's kind. They were just more advanced, as time had gone on and they were in less forbidding lands. An ancient Cro-Magnon Man, for example, would be different from one of us; yet he would be very similar, as he is the same species. He is also, if you are white and European, the same race. That, I believe, the relationship between Gollum's comparatively medieaval Stoor society and the Shire society of those Stoors who lived there. If Gandalf didn't know about them, it's because they were well hidden, small in population, secretive (like all Hobbits) and insignificant in the scale of things. They were also a long time ago, and Gandalf himself was new to Middle-Earth at that stage as well.