If you need to do this in a module or a larger program, you can localize the filehandle glob, or use the two-argument select. This is handy to avoid spooky action at a distance:

open(OUT, ">firstout.txt") or die "1: $!\n";
print OUT "First\n";

{
    local *OUT;
    open(OUT, ">secondout.txt") or die "2: $!\n";
    print OUT "Second\n";
}

print OUT "Third\n";
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Update: If there's an explicit close, it's also worth checking for an error. Someone on p5p had intermittent test failures (creating temporary files) on a nearly full disk. I rarely check, but know I should.