Are you sure?

At my previous company (a relatively small startup, admittedly), one of the indisputably best engineers (a 10xer if I've ever worked with one) was hired after such a "second shot". A few weeks is probably too rapid a turn around, given that a "failed" interview is a negative signal (or at least, should be That's the point, after all). But say, a year later for engineers in that wide fuzzy area between "Hire!" and "Oh god no!"? I think plenty of companies do do that.

Further anecdata: Friends who've been rejected from Google have told me that they were explicitly encouraged by their Google recruiter to re-apply in ~a year.