Well in the current situation, it is very unlikely you will get a good one in a short time. Usually, the starting day is negotiable. Most labs can wiat half a year for your graduation (after you sign the contract). 又或 You need to contact several of them on the sametime. And see if you can get a nice offer. If you cannot. You can continue to search. Searching for postdoc is not like applying for school because there is no fixed deadline. Of course yes !!!!