If I understood correctly you are trying to use a single query, i think you can just merge your date search toghter in WHERE
clauses
SELECT id
FROM events
WHERE start BETWEEN '2013-06-13' AND '2013-07-22'
AND end BETWEEN '2013-06-13' AND '2013-07-22'
or even more simply you can just use both column to set search time filter
SELECT id
FROM events
WHERE start >= '2013-07-22' AND end <= '2013-06-13'