$query->set
isn’t the only method.
What you want is $query->get
. Using get
you can retrieve your existing value, append the additional part you wanted, then set the query variable with the new argument
$tax_query = $query->get( 'tax_query' );
// add the extra parameter
$query->set( 'tax_query', $tax_query );
The adding the extra parameter is just appending another item using standard PHP array functionality.