Let's say I have a couple of hashes that I need to bruteforce as dictionary attack didn't work. Is there a way I can tell hashcat to start from a specific password length so it won't waste resources on looking for results that won't work?
Sounds like you're looking for
--increment-min. This will start a bruteforce/mask attack at a minimum length.
For example, this will try digits-only candidates, starting with length 7:
hashcat -a 3 -m [hashtype] -i --increment-min=7 targethashes.list ?d?d?d?d?d?d?d?d?d?d
You didn't directly ask this, but the deeper intent of your question appears to be "what can I do beyond a dictionary attack?"
There are many other options to try before you go full bruteforce - combinator, hybrid, masks. See hashcat's core attack modes for some introductory materials.