"Yes, I had already supposed it."
Translation:Sí, ya me lo había supuesto.
This construction occurs elsewhere in this section of the tree. See
Había can be I, he, she, it. The "me" shows which is the subject.
There is an alternative explanation offered by some people which focusses on the way in which many verbs such as 'tomar' and 'suponer' change meaning in the reflexive.
Still others would use "yo" instead of "me" or omit it altogether.
I have a feeing this may be a regional thing. I will try to ask a couple of local speakers here in Gran Canaria and add to this comment if I find anything definitive.
This site says: suponer can mean suppose, imagine, mean (imply), whereas suponerse can mean assume, imagine, suppose. The site gives examples that are helpful.