It seems to me like you should make a hash of arrays mapping dates to their records. So, you could do something like this:

my %records;
while( my $row = $sth->fetchrow_hashref ) { 
    push @{ $records{ $row->{start_date} } }, $row;
}
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Then you can loop through that hash (sorting as necessary) to output the records. Since the outer hash is keyed by the datestamp, we can sort it easily:

foreach my $date( sort keys %records ) { 
    print $date, "\n";

    # then get each record, sorted by the start time
    my @recs = sort {
        $a->{start_time} cmp $b->{start_time}
    } @{ $records{$date} };

    foreach my $rec( @recs ) { 
        print $rec->{id_number}, "\t", $rec->{start_time};
        print "\n";
    }
}
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That's all untested but should get you on the right track.