She moved to our school in the middle of her first year in high school. That following summer, she asked the coach if she could join the team as a sophomore.
1) What's the meaning difference between 'that following summer' and 'the following summer'? Especially, what meaning does "that" add?
2) What part of speech is 'that' in the "that following summer"? a determiner as in [[that][following]summer] or an adverb as in [[that following] summer]?