After taking a look at this SO question and doing my own research, it appears that you cannot have a leading wildcard while using full text search.

So in the most simple example, if I have a Table with 1 column like below:

TABLE1

coin
coinage
undercoin

select COLUMN1 from TABLE1 where COLUMN1 LIKE '%coin%' Would get me the results I want.

How can I get the exact same results with FULL TEXT SEARCH enabled on the column?

The following two queries return the exact same data, which is not exactly what I want.

SELECT COLUMN1 FROM TABLE1 WHERE CONTAINS(COLUMN1, '"coin*"')

SELECT COLUMN1 FROM TABLE1 WHERE CONTAINS(COLUMN1, '"*coin*"')