Is this the best performance using mysql count on multiple joins - mysql

Question

I have a DB with the following tables:
employee
company
lt_employee_title
Employees can have multiple titles and companies can have multiple employees.
I'm trying to search on the lt_employee_title table and the company table and get employee_id from the employee table.
The table structures:
CREATE TABLE `employee` (
`employee_id` int(11) NOT NULL AUTO_INCREMENT,
`company_id` int(11) DEFAULT NULL,
PRIMARY KEY (`employee_id`),
KEY `employee_id` (`company_id`)
) ENGINE=InnoDB;
CREATE TABLE `company` (
`company_id` int(11) NOT NULL AUTO_INCREMENT,
`URL` varchar(100) DEFAULT NULL,
`company_name` varchar(100) DEFAULT NULL,
`date_created` date DEFAULT NULL,
`date_updated` datetime DEFAULT NULL,
`address1` varchar(45) DEFAULT NULL,
`address2` varchar(45) DEFAULT NULL,
`phone` varchar(20) DEFAULT NULL,
`phone2` varchar(20) DEFAULT NULL,
`fax` varchar(20) DEFAULT NULL,
`city` varchar(45) DEFAULT NULL,
`major_city_id` int(11) DEFAULT NULL,
`region_code` varchar(10) DEFAULT NULL,
`zip` varchar(15) DEFAULT NULL,
`country_code` varchar(2) DEFAULT NULL,
`archived` tinyint(4) NOT NULL DEFAULT '0',
`enabled` enum('yes','no') DEFAULT 'yes',
PRIMARY KEY (`company_id`),
KEY `country_code` (`country_code`),
KEY `employee` (`number_of_employees_id`),
KEY `phone` (`phone`(3)),
KEY `state` (`region_code`),
KEY `archived` (`archived`)
) ENGINE=InnoDB;
CREATE TABLE `lt_employee_title` (
`employee_id` int(11) NOT NULL,
`title_id` int(11) NOT NULL,
UNIQUE KEY `unique_field` (`employee_id`,`title_id`),
KEY `employee_id` (`employee_id`),
KEY `title_id` (`title_id`),
KEY `both` (`title_id`,`employee_id`),
KEY `both2` (`employee_id`,`title_id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8$$;
This query is fast:
SELECT SQL_NO_CACHE
*
FROM lt_employee_title AS let
JOIN employee AS ee
ON ee.employee_id = let.employee_id
JOIN company AS com
ON com.company_id = ee.company_id
WHERE let.title_id = 9
AND com.region_code IN('AL', 'AK', 'AZ')
AND com.archived = 0
LIMIT 125;
But these are not:
SELECT SQL_NO_CACHE
COUNT(*)
FROM lt_employee_title AS let
...
SELECT SQL_NO_CACHE SQL_CALC_FOUND_ROWS
*
FROM lt_employee_title AS let
...
The difference of adding any type of count moves the time from 0.016 seconds to 4.0 seconds. We have about 250,000 titles, and 900,000 companiy entries and over 1 million employee records but will be growing.
When I add in count() or SQL_CALC_FOUND_ROWS or ORDER BY com.number_of_employees_id, the EXPLAIN is using temp tables and file_sort which is why the speed decrease. Is there a way to do this sql more efficiently?
EXPLAIN WITH "SQL_CALC_FOUND_ROWS"
1 SIMPLE company ref PRIMARY,state,created,updated,employee_number,archived state 33 const 210864 Using where; Using filesort
1 SIMPLE employee ref PRIMARY,employee_id,created,updated employee_id 5 lead411_main.company.company_id 1 Using where; Using index
1 SIMPLE lt_employee_title eq_ref unique_field,employee_id,title_id,both,both2 unique_field 8 lead411_main.employee.employee_id,const 1 Using index
EXPLAIN WITH OUT "SQL_CALC_FOUND_ROWS"
1 SIMPLE company index PRIMARY,state,created,updated,employee_number,archived employee 5 986 Using where
1 SIMPLE employee ref PRIMARY,employee_id,created,updated employee_id 5 lead411_main.company.company_id 1 Using where; Using index
1 SIMPLE lt_employee_title eq_ref unique_field,employee_id,title_id,both,both2 unique_field 8 lead411_main.employee.employee_id,const 1 Using index