Query optimizer doesn't use best index even with INDEX hint.

I am running database 11g version 11.2.0.4. I have a table TRN_Membership with a bunch of fields including two varchar fields named Organization and Last_First. There is a normal index named trn_membership_name_index on organization, last_first.

I would expect the query:

select * from trn_membership where organization='usta' and last_first = 'P'

to use the name_index rather than a full scan and it does, but if I change the query to

select * from trn_membership where organization='usta' and last_first >= 'P'

it switches to a full access scan. I added the hint /*+ INDEX (trn_membership) */ to force the use of an index but it does not change the sql plan. I even was explicit with the hint /*+ INDEX (trn_membership trn_membership_name_index) */ but the optimizer would still not do an index scan.