Make sure the filename is based on the current date and always open for each write. Something like

sub loggit {
    my @gmt  = gmtime;
    my $date = sprintf "%04d%02d%02d", $gmt[5]+1900, $gmt[4]+1, $gmt[3
+];
    my $file = "$date.log";

    open my $log, ">>", $file or die $!;
    print $log, @_;
    close $log;
}
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