wood is fine but make sure to waterproof it if used outside. use as little as practical, performance can change with weather unlike with plastic. no metal fasteners either as this will have weird effects
do you want it to be compact when deployed or compact when stored? you can also make crossed dipole yagi, you probably want it to be rather nondirectional so one director, driven element and reflector would be fine. that antenna is much longer but i think you get the idea https://www.qsl.net/dk7zb/Cross-Yagi/crossyagi.htm you can also make bifiliar helical antenna but this one again gets a bit directional. you can probably make it shorter than these 5 turns and change pitch too to decrease directionality https://secwww.jhuapl.edu/techdigest/content/techdigest/pdf/V12-N01/12-01-Stilwell.pdf you can also make unifilar helical antenna but it needs a big reflector baseplate so it won't be as compact, it looks like this https://www.ab9il.net/wlan-projects/wifi3A.html