Baumann, Brewka and Ulbricht recently introduced weak admissibility as an alternative to Dung’s notion of admissibility, and they use it to define weakly preferred, weakly complete and weakly grounded semantics of argumentation frameworks. In earlier work we introduced two variants of their new semantics, which we called qualified and semi-qualified semantics, and we analyzed all known variants of weak admissibility semantics with respect to some of the principles discussed in the literature on abstract argumentation, as well as with respect to some new principles we introduced to distinguish all of them. Besides selecting a semantics for an application, or for algorithmic design, such a principle-based analysis can also be used for the further search for weak admissibility semantics. In this paper we introduce six new kinds of semantics based on weak admissibility, and we provide an initial principle-based analysis. The analysis illustrates various ways in which the new semantics improve on the existing ones.