I’m assuming this is wind damage and not flood / water damage.  

If wind damage, then your Homeowner’s Insurance would be the first contact.  You can contact either your policy claim department if you have your paperwork, or you can call your insurance agent / office for who you need to speak with.  Not that your insurance agent / office may also be damaged and not readily answering the phones for a few days if they are local to the area.  Earlier is likely better than later to get the insurance adjuster scheduled.

Public adjusters can be useful, however since they charge a significant chunk of the eventual claim for their services, it usually only makes sense if there is significant damage.  

When I was flooded, it didn’t really make sense unless the claim was at least $80k – $100k+ range since their fee was more than any potential increase in the claims amount.  

However, my experience was with a FEMA flood insurance claim.   Homeowner’s Insurance is an entirely different (and usually more pleasant experience) for things like wind or tree damage. So you will need to evaluate for your specific situation.