I can understand hitting Google or doing a facebook search, out of curiosity or a yearning for connection... but, ultimately, I don't think much good ever comes out of it.

Attempting further methods of obtaining information ...?

What are you trying to find out, and why? I'm not judging, but I think it could be more valuable to explore the reason behind the motivation than to keep digging for scraps of information that could potentially open cans of worms. Oh, the cans of worms!! At best, if you do find anything out, you could end up feeling guilty over how you found what you did, and then the uncomfortable position of pretending you don't know! Or maybe you find something that makes you uncomfortable, or jealous.. or makes you question something.. and again, you're not in the position to talk about it unless you reveal what you did, which could potentially cause issues in of itself.

Another method for finding things out about T might be to ask her what you'd like to know? A lot of Ts will self disclose to some extent - if they feel it's relevant, will be therapeutic, or will help to form a bond and/or build the relationship/trust. I would aim to keep questions respectful, though, and not too personal in nature (though where that line is will likely vary depending on the T)... and keeping in mind that she may decline to give you the answers - although it could be a valuable conversation all the same..