Create multiple field indexes with different types on postgres
This is going to be impossible for us to assist in with the information provided. However, rest assured 18M is not much. Baseline questions include,


Ignoring the LIKE condition, what is the selectivity that can be provided on start_time and product?
Do you ever select on start_time, and name, without product?
When you have a btree on (start_time, product), and a gin/gist tgrm index on name, how much times does the query take?
When you create the index described, how much time does the query take? 


Take a look at the plans required by both. I would do these tests on a snap shot of the table so it's fair (or two snapshots with only the indexes mentioned)

CREATE TEMP TABLE foo
AS
  SELECT * FROM marketing