There is now a 60-day grace period after the end of H-1B employment so you would be considered in valid status and "portable" to a new company until the end of

There is now a 60-day grace period after the end of H-1B employment so you would be considered in valid status and "portable" to a new company until the end of February, meaning that as long as the new employer files the H-1B petition before the end of February, you can remain in the U.S. while the petition is pending and even commence employment upon the receipt of the petition. And yes, different companies can file H-1B petitions at the same time but this might cause USCIS to question - if it is made aware of this - the bona fides of the various job offers.