I believe you have misunderstood the use of $ENV{'PATH'}. It is the path searched to find executables by name when no full path is given. When found, the executable runs in $ENV{'PWD'}, the current working directory.

gzip will happily do what you want if you just tell it where to find the file. Adjusting your pseudocode:

while (args here) {
    $new_dir = code to figure out correct path...
    $file = code to determine file to work on...
    $fullpath = "/home/mysite/documents/daily/$new_dir/$file";
    system("gzip $fullpath") == 0 or die "can't gzip file";
}
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After Compline,
Zaxo