Is it permissible for a pilgrim performing Tamattu' Hajj, after exiting the state of ihram from 'Umrah and before entering the state of ihram for Hajj, to perform another 'Umrah for himself or for one of his relatives?

Repeating 'Umrah is supererogatory, whether it's after the 'Umrah of Tamattu' or after Hajj. The evidence for this is the Prophet's (peace be upon him) saying: "From one 'Umrah to the next is an expiation for what is between them, and Hajj Mabrur (an accepted Hajj) has no reward but Paradise." This is the view of the majority of scholars. There is no dislike in performing it frequently in a single year or a single day; rather, it is supererogatory without dispute.