"That" between "thought" and "could"

It was me that Mr. Jones thought could do it.

correct?

Is that required between thought and could?

Solution 1:

I always thought only the references to inanimate objects use that or which, otherwise who should be used.

"It was me who Mr. Jones thought could do it." - sounds perfectly fine to me. ("Mr. Jones thought I could do it.")

P.S. I intended the text as a comment to an answer above, but cannot comment yet due to the reputation level.

With a few more changes I think this is even more natural:

I was the one who Mr. Jones thought could do it.