**Revised**: The proposed construction is just fine, and in particular:

- at least as secure as SHA-256 against collision attacks, that is the ability for an adversary to construct two files with the same hash;
- likely about as secure as SHA-256 against both first and second preimage attacks, that is the ability for an adversary to construct (for first preimage) a file with some hash given as an arbitrary value, or (for second preimage) a file with the same hash as an arbitrary given file.

The construction would slightly reduce the second-preimage resistance of a maximally resistant hash. But for SHA-256, the second-preimage resistance seems to remain no worse than allowed by a generic attack on Merkle-Damgård hashes attributed to R. D. Dean in his 1999 [thesis][1] (section 5.3.1), better exposed and refined by J. Kelsey and B. Schneier in [_Second Preimages on $n$-bit Hash Functions for Much Less than $2^n$ Work_][2].

  [1]: http://www.windowsecurity.com/uplarticle/10/ddean-thesis.pdf
  [2]: http://www.schneier.com/paper-preimages.html