Reusing covering index for fields in FROM, from another query's index

For the following query I have a covering index as so:

SELECT bar_id FROM foo WHERE user_id=:user_id AND date_sent=:date_sent;

ALTER TABLE foo ADD INDEX (user_id, date_sent, bar_id);

Now I have another query that I would like to add a covering index for:

SELECT user_id FROM foo WHERE bar_id=:bar_id AND date_sent=:date_sent;

Does the previous covering index also cover this new query? When I run EXPLAIN for the new query the following output is returned:

+----+-------------+-------+------+---------------+--------+---------+-------+------+-------+
| id | select_type | table | type | possible_keys | key    | key_len | ref   | rows | Extra |
+----+-------------+-------+------+---------------+--------+---------+-------+------+-------+
|  1 | SIMPLE      | foo   | ref  | bar_id        | bar_id | 4       | const |    5 | NULL  |
+----+-------------+-------+------+---------------+--------+---------+-------+------+-------+

I see that the key mentioned is bar_id, but that seems to me to show that this is not a covering index. Furthermore, the Extra field does not show the expected Using index output that would assure me that a covering index is in use.