My understanding is that a hypothetical syllogism is in which there are three statements in which the premises and the conclusion are all hypothetical in form. Classically :
((A => B) ∧ (B => C)) ⊃ (A => C).
Your example fits this. Here's another, just to show a variation |:
If P is true, Q is true
If R is true, P is true
therefore, If R is true, Q is true
The traditional 'Scholastic' logic of the Middle Ages drew distinctions between conditional hypotheticals, disjunctive hypotheticals, and conjunctive hypotheticals. All very interesting but the examples above, yours and mine, illustrate the simple hypothetical of your question.
In both examples the conclusion follows logically from the premises. It's possible, of course, to produce hypothetical syllogisms which go wrong and in which the conclusion does not follow from the premises. We try to avoid these ;)-