Suppose that i've a table (t1) with 3 columns (c1,c2,c3), and i added an index like this:
ALTER TABLE `t1` ADD INDEX `IDX_c1_c2` USING BTREE (`c1`, `c2`);
SELECT * FROM `t1` where c1 = "foo" AND c2 = "bar"
SELECT * FROM `t1` where c1 = "foo" AND c2 = "bar" AND c3 = "foobar"
SELECT * FROM `t1` where c1 = "foo" AND c2 = "bar" AND c3 > "2016-11-26 07:37:47"
Just to make things simple your exisiting index will still be used , however third column will be resolved independently. The Query Processor will generate some plans and the best plan will be accepted by the engine to proceed . Still making third column an index as well will definitely boast the performance .
Suggestion If you are using equality conditions in querying go for hash based indexing rather than Btree