If you would have used an OO framework like CGI::Application for the structure of your app, you would have been able to do this very easily, without needing to subclass HTML::Template.

package My::App;

use base qw(CGI::Application);

sub load_tmpl {
  my $self = shift;

  my $template = $self->SUPER::load_tmpl(@_);
  # add some global params here
  $template->param( ... );
  return $template;
}

sub runmode {
  my $self = shift;
  my $template = $self->load_tmpl('filename.tmpl');
  $template->param( ... );
  return $template->output;
}
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OO doesn't work for everything, but if you find yourself passing the same bits of data to all of your functions, then you probably should at least investigate using an object instead.