Relative Clauses- To use or not to use That

Solution 1:

The omission of the objective relative pronoun results in what is sometimes called a contact clause (also zero clause). There is a good entry about contact clauses on ThoughtCo.

There is no rule that will help you determine whether to omit that or not. It is a stylistic choice. Merriam–Webster's Dictionary of English Usage (p290) claims:

Constructions such as these are appropriate in any variety of writing.

The Cambridge Guide to English Usage (p468) is a little more cautious:

Except in most formal style, the pronoun can be omitted from relative clauses of which it is the object.

So, if you are aiming at a high level of formality you may wish to retain that.