In your current implementation you could (ab)use the EXPORTS_OK list to import the constants as template variables:

my $content_ref = { map { $_ => Sam::Constants->$_ } @Sam::Constants::EXPORT_OK }

You could also pre-load them into the configuration of your >TT object. Here's a complete example:

use strict;
use warnings;
package Sam::Constants;

use base 'Exporter';

use constant PRE     => 1;
use constant REGULAR => 2;
use constant POST    => 3;

our @EXPORT_OK = qw( PRE REGULAR POST );

package main;
use Template;

my $text = <<EOT;
The constant 'PRE'     has the value [% PRE %]
The constant 'REGULAR' has the value [% REGULAR %]
The constant 'POST'    has the value [% POST %]
EOT

my $tt = Template->new(
    { VARIABLES =>
      { map { $_ => Sam::Constants->$_ } @Sam::Constants::EXPORT_OK },
    }
) || die "$Template::ERROR\n";
$tt->process( \$text )
    || die $tt->error(), "\n";
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When run, this prints:

The constant 'PRE'     has the value 1
The constant 'REGULAR' has the value 2
The constant 'POST'    has the value 3
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