What type of index do I need for this JOIN?

I'm doing a JOIN which is very slow:

JOIN t ON t.f1 LIKE (q.f1 || '%')

t1 has an INDEX on (f1, f2) which I thought would help for this.  But
Postgres seems to still use a (very slow) Nested Loop.  What type of
index would be appropriate for this?

(My goal is to join on a substring starting from the first character)