If you find such a set then it cannot be stricly greater than the cardinality of the integers and strictly less than the cardinality of the continuum, because as you said this would violate the Countinuum hypothesis thus we have two possiblities.

1)The continuum hypothesis is false and the cardinality of the continuum is NOT the minimal uncountable set.

2)The continuum hypothesis is true then the minimal cardinality must be the the set of the continuum.

But as I understand this topic correctly there is NO ANSWER TO YOUR QUESTION

Because (and I might be wrong on this) the continuum hypothesis is independent from ZFC (even with the Axiom of Choice). Thus, we cannot conclude it is either true or false.

I think the answer to your question is the continuum.