You should be able to claim vested SAFE tokens in the Safe app as long as you claimed any portion of the original SAFE issuance before the first deadline.
The redistribution of unclaimed SAFE tokens contract will need to be drafted and implemented first. It seems that the July 1st, 2023 date you’re referring to is the date mentioned in a draft to use an existing airdrop contract and has not been implemented. There could be a deadline to claim the redistributed tokens like in the original token issuance or the contract could implement no claim deadline.