Here's an example of how to accomplish this:

use List::Util qw/min max/;

my %results = map {
    my $year_ref = $data{$_};
    my $min = min values %$year_ref;
    my $max = max values %$year_ref;
    my @min_months = grep { $year_ref->{$_} == $min } keys %$year_ref;
    my @max_months = grep { $year_ref->{$_} == $max } keys %$year_ref;

    $_ => {
        "min"           => $min,
        "max"           => $max,
        "min_months"    => \@min_months,
        "max_months"    => \@max_months
    }
} keys %data;
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Note that this takes into account the possibility of there being several months that each have the minimum/maximum value; if your data for each year/month is guaranteed to be distinct, you could drop that and replace \@min_months with $min_months[0] and \@max_months with $max_months[0], respectively.

There may well be a more concise way of doing this, too, but neither List::Util nor List::MoreUtils appears to contain versions of min and max that operate on key/value pairs (pairmin and pairmax? Now there'd be a worthwhile addition to these modules.)