My solution to Problem 4 in Problem Set 2 is an adaptation of Hirsch'98. This solution is somewhat dissimilar to Sipser-Saxe-Furst (contrary to what the hint in the Problem Set suggests) and so I wonder if someone has a different (easier) solution.
On the other hand, reading Hirsch's paper raised an interesting question. As a corollary of this exercise we know that formulas with bounded-width clauses and many satisfying assignments have short (constant size) partial satisfying assignments. It is an open question (according to Hirsch) whether a similar procedure exists for CNF formulas with unbounded-width clauses.
To suggest that maybe this is not possible, (in Remark 6 of Hirsch'98) Hirsch constructs a CNF for which more than half of all assignments are satisfactory and yet for every partial assignment on SQRT(n) variables there exists a corresponding complete assignment which is unsatisying.
Hirsch, however, fails to point out that the constructed CNF formula might (and most likely will) be exponential in size. This raises a new, more concrete, open question: Is there a way to modify the present algorithm such that it takes advantage of bounds on formula length in addition to clause-width that solves the problem for CNFs with unbounded-width clauses in sub-exponential time?