Once the limbs dry out it becomes very difficult to stripe the bark from the walking stick making wood you collect. If you plan to leave the bark on to carve then you can until later Fall and even into winter before the trees get their first coating of snow/ice.
Many people ask about exotic woods and tropical woods to make hiking sticks. I live in south Florida where popular, aspen, willows and other woods for stick making are just dreams. I have experimented with various tropical woods and a very few are good candidates and most are poor candidates for hiking stick making projects. Flamboyant is a good one but stringy. Bottlebrush trees used in landing scaping here works well but is extremely hard wood. Scrub oak is a poor choice unless you get it young. One f my favorites is cypress saplings which when dry extremely light but are very strong (supports weight).
There are several different types of cypress and make sure you do not collect walking stick making wood from the everglades or other parks. It's best to see where building are clearing land and ask if you can get some of the limbs/sapling they are digging up to clear the land. One plant that is a nuisance in south Florida is the melateuca tree. It's an invasive tree from Malaysia over 100 years ago to drain water form the land - needles to say this is a tree you can have as much as you want. It bark is lik paper and makes interesting walking sticks.