Beyond the assignment to $opt{input}, your requirement seems to be for the normal operaton of Getopt::Long.

Consider this code:

#!/usr/bin/env perl

use 5.010;
use strict;
use warnings;

use Getopt::Long;

my %opt;

GetOptions(\%opt, 'v');

say 'Before assigning $opt{input}:';

if ($opt{v}) {
    say $ARGV[0], ' (verbose)';
}
else {
    say $ARGV[0], ' (quiet)';
}

$opt{input} = $ARGV[0];

say 'After assigning $opt{input}:';

if ($opt{v}) {
    say $opt{input}, ' (verbose)';
}
else {
    say $opt{input}, ' (quiet)';
}
[download]

Using your two command line examples, this produces:

$ pm_getopt_arg.pl ex-post-facto
Before assigning $opt{input}:
ex-post-facto (quiet)
After assigning $opt{input}:
ex-post-facto (quiet)
[download]

$ pm_getopt_arg.pl -v ex-post-facto
Before assigning $opt{input}:
ex-post-facto (verbose)
After assigning $opt{input}:
ex-post-facto (verbose)
[download]