A hash slice solves the problem (see perldata - slice):

use strict;
use warnings;

my %teamStats;
my @paramNames = ('points', 'tallies', 'scores', 'metrics');

while (<DATA>) {
    chomp;
    my ($team, @params) = split /\s+/;
    
    next if @params < 4;
    
    # Populate new hash using a hash slice
    my %newParams;
    @newParams{@paramNames} = @params;
    push @{$teamStats{$team}}, \%newParams;
}

foreach my $team (keys %teamStats) {
    print "\n\nTeam:\t$team\n";
    print "\t", join ("\t", @paramNames), "\n";
    
    for my $game (@{$teamStats{$team}}) {
        # Hash slice to the rescue again
        print "\t", join ("\t", @$game{@paramNames}), "\n";
    }
}


__DATA__
packers    4    40    400    4000
patriots    6    62    436    3987
colts    8    74    892    7666
bears    9    88    912    9550
packers    3    44    410    4200
patriots    7    66    510    3800
colts    10    77    910    7000
bears    11    88    1010    9410
[download]

Prints:

Team:    colts
    points    tallies    scores    metrics
    8    74    892    7666
    10    77    910    7000


Team:    patriots
    points    tallies    scores    metrics
    6    62    436    3987
    7    66    510    3800


Team:    packers
    points    tallies    scores    metrics
    4    40    400    4000
    3    44    410    4200


Team:    bears
    points    tallies    scores    metrics
    9    88    912    9550
    11    88    1010    9410
[download]

Update: cleaned up a little "legacy" code