Clustered Table Scan Because of "SELECT *"
If you need columns in the output that aren't covered by the index, the optimizer has to make a choice:


Perform a table / clustered index scan (therefore all columns are there)
Perform a seek, then perform lookups to retrieve the columns not covered


Which way it will choose depends on a variety of things, including how narrow the index is, how many rows match the predicate, etc. You can force a seek with the FORCESEEK hint, but I suspect it will end up performing the same or worse than the scan SQL Server has chosen in your case.

Some options:


Change the app to run a proper query. I listed this first for a reason.
Create a view that selects only the columns you need:

CREATE VIEW dbo.myview
WITH SCHEMABINDING
AS
  SELECT col1, col2, col3 FROM dbo.tablename;


Then you can change the app to SELECT * from this view. Or you can get even more creative and rename the original table, and change the name of this view to what the name of the table used to be. Breaking change, obviously; proceed with caution.
Add all of the other columns to the key or INCLUDE list for the index. If these are hard-coded values and always the ones used, you may consider a filtered index.