I suggest a different approach...

note: I have not tested this code. The concept has worked elsewhere, but there may be typos or incompatibility with File::Tail.

#!/usr/bin/perl
use strict;
use warnings;

use File::Tail;

my $cur_filename = $ARGV[0];

{
    my $file = File::Tail->new(
        name    => $cur_filename,
        maxinterval => 3,
        adjustafter => 2,
        tail    => -1,
    );

    eval {
        local $SIG{INT} = sub { die "interrupted\n"; };

        while ( defined( my $line = $file->read) ) {
            print $line;
        }

    };
    
    if($@ eq "interrupted\n") {
        open(FILE,"/tmp/filename") || die "Cannot open file:$!";
        $cur_filename=<FILE>;
        close(FILE);
        redo;
    } else {
        die "$@";
    }
}
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You may never print the last line of the file if the interrupt arrives after a line has been added and before it has been printed: an interval including a significant number of CPU cycles. This risk can be mitigated somewhat by having the external program redirect log entries to the new file then waiting briefly before interrupting this program.