I have recently got to know these notes from this answer, i.e.,
Will Merry, Differential Geometry: beautifully written notes (with problems sheets!), where lectures 1-27 cover pretty much the same stuff as the above book of Jeffrey Lee. Lectures 28-53 also center around metrics and connections, but the notion of parallel transport is worked out much more thoroughly than in Jeffrey Lee's book.
It's unfortunate that the author passed away and the link (to the notes given in that answer) does not work anymore.
I would like to ask if someone keeps these notes and can share them. Thank you so much!