Nef vector bundles on a projective space or a hyperquadric with the first Chern class small

We give a new proof of the classification due to Peternell–Szurek–Wiśniewski of nef vector bundles on a projective space with the first Chern class less than three and on a smooth hyperquadric with the first Chern class less than two over an algebraically closed field of characteristic zero. We also classify nef vector bundles containing a line bundle of next-to-maximal degree on a projective space.