I was about to ask the exact same question now. Hirsch's book drew me in with its table of contents but it turns out his proof style is very crude. At one point instead of defining partitions of unity subordinate to covers he maps the different charts to balls of different sizes in $\marhbb{R}^n$ and then uses a cut off function on the balls. VERY confusing! Anyway if anyone has a different source for the same material that appears in -Hirsch I could really use the recommendation.

Yes, I have two accounts since i was too shy to ask a certain question and wanted to do it anonymously. Now, thanks to my recklessness my cover is blown...