Classical rites that mention dew in the second blessing consistently also request it in the ninth blessing:
- Sefardim say מוריד הטל and וברך שנתנו בטללי רצון
- Italians say מוריד הטל and ותן טל לברכה
- Yemenites say מוריד הטל and וברך את שנותינו בטללי רצון.
Those that don't mention dew in the second blessing also don't request it in the ninth blessing:
Tosfot (Taanit 3b) does quote one rabbi who would mention dew in the second blessing all year so as never to accidentally need to go back, though he never indicates he thought that was a traditional practice.
This is all consistent with the Talmud (Taanit 3) that mentioning dew is completely optional. You could choose to mention dew in either, both or neither of those blessings at any time of year and it wouldn't harm your prayer at all.