I have never heard the saying to put someone on to someone else. I would say: We can put you in touch with her.

Your guess is right: to refer someone to someone else/put them in contact/give them details or information. The phrasal verb is: To put someone on to (onto) someone or something. The phrasal verb can be used for people (as in your example) or for things. "She put me on to a really good textbook that explains everything." Other examples (and places to be careful!): My doctor put me on TO a good cough remedy. (This is a little ambiguous. Either it means the doctor told me about it but did not supply it, or it might mean prescribed it!). (Care!) My doctor put me ON a good cough remedy. (The doctor prescribed the medicine) My friend put me on to an excellent diet. (Told me about it) My friend put me on an excellent diet. (My friend made me go on a diet!) The government advice centre can put you on to all sorts of helpful people. They will put you on to the right person, if you call them. And finally, don't confuse it with 'putting someone on' - as in transferring a phone call to them. "I'll put her on." (yes, I know 'putting someone on' has another meaning - but enough 'put' for today!)

Thanks for your addition, Faye.

Yes you guessed correctly. People sometimes say things in odd ways. There is slang too that is very close to this “put me on with her” “put you on with her” this means help you date or hook up with someone. It’s mostly used by young people online, when a girl (or boy) introduces a friend or posts a photo with someone else, someone may find that person attractive or interesting, and they say “put me on “ or “put me on with him/her”