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!)