I found this in the PHP manual (Chapter 19. Handling file uploads):
Files will by default be stored in the server's default temporary directory, unless another location has been given with the upload_tmp_dir directive in php.ini. The server's default directory can be changed by setting the environment variable TMPDIR in the environment in which PHP runs. Setting it using putenv() from within a PHP script will not work. This environment variable can also be used to make sure that other operations are working on uploaded files, as well.
I wonder if TMPDIR could be changed in an .htaccess file.