Minimizing catastrophic risk in Gale-Shapley matching
(In theory) No, no other strategy will help. In the student-proposing version of the deferred acceptance algorithm, it is a theorem[0] that each student's optimal action is to report her true preference order. It's a dominant strategy, meaning this is optimal regardless of what others submit. This implies that if the student is unmatched when submitting her true preferences, then there is no other submission she could make that would have gotten her matched[1].

Note deferred acceptance still works for weak preference orderings (see [0]), so e.g. here the student is indifferent between all hospitals and prefers them all to being unmatched. The dominant-strategy guarantee of deferred acceptance still holds. 

(In practice) In practice, students don't interview or rank all possible hospitals. So a student could likely improve her chances by trying to interview and rank mostly lower-prestige programs where the competition is lower.

[0] e.g. Theorem 8 of the survey by Al Roth, Deferred Acceptance Algorithms: History, Theory, Practice, and Open Questions. https://web.stanford.edu/~alroth/papers/GaleandShapley.revised.IJGT.pdf

[1] I am assuming she prefers any match to being unmatched.