Suppose that we start out with the pair of concepts substance and property so that we can generically say things like:
- X is a substance.
- X is a property.
- X is both a substance and a property.
- X is neither a substance nor a property.
An intermediary between, or supersession of, substance- and property-dualism might be cashed out in terms of (3) or (4), then. However, we should also be inclined to consider variations on the concept of substance or on the concept of properties. In bundle theories of object-talk, for example, the concept of a substance to which properties adhere/attach is replaced by the concept of a set of properties, where the principle for identifying which elements are in such a set (and hence which characterize separate objects when we have separate sets) is not substance-theoretic. Perhaps, then, one might identify a mind/matter dualism where mind is somehow equated with a principle of bundling, perhaps as a "meta-property" of the properties in some bundles. (I am generally sure that there are essays out there in which such proposals are explored, but I don't have any specific knowledge of such essays.)
Another relevant ambiguation over property theory could involve Zalta's distinction between encoding and exemplifying properties. Normal (property) dualism could be taken as the claim that there can be mental properties which are exemplified by non-mental particulars. But one might try imagining that mental properties are never exemplified by non-mental particulars but are possibly encoded by/in them (or that mental properties are sometimes non-mentally exemplary and sometimes non-mentally encoded, etc.).
Then one could go on to imagine theories where mental substances either only exemplify or only encode mental properties, or where mental substances can satisfy both property relations, where there are mental properties exemplifiable by mental substances and encoded by non-mental ones, and so on and on.
With respect to the OP subquestion "mental states continuing after death": so then suppose that we are physical substances which can variously encode and exemplify mental properties. Lindsky and Zalta pose the question of contingently abstract objects and there are alternatives to property theory, known as trope theories, which involve a question of abstract particulars. So we could imagine that there are contingently abstract mental particulars, such that relative to a physically living person these particulars are concrete, but so that relative to such a person after death (as well as before life), these particulars are abstract.
Such an account is (roughly) similar to the Catholic doctrine of souls as "forms" of human bodies, where "forms of" is read in an Aristotelian manner. (C.f. the concept of haecceities, which might be thought to exist necessarily such that, albeit indexed to exact particulars, their existence is independent on the occurrent existence of those particulars. One philosophically self-aware fiction writer nowadays, Brandon Sanderson, appears to define something specter- or spirit-like about physical beings in such terms.)
ADDENDUM: Modality-theoretic options
At a squiggly diagonal off from the above-mentioned haecceity-talk, there is a debate in the theory of modality between actualists and possibilists, which is roughly a debate over the existence of "mere possibilia" (things that are possible simpliciter, rather than possible only in relation to actualities). So one might imagine a dualism such that actual substances are strictly physical whereas mental substances are strictly merely possible when further mental properties are not physically instantiated (as actualities), etc.