"Voting systems" are mechanisms, but we also design mechanisms for situations that aren't usefully construed as voting; in terms of practically used mechanisms, I'm thinking especially of auctions and other allocation and matching mechanisms. Typically these allocation mechanisms try to optimize the outcome in some sense; where centralized matching mechanisms have replaced decentralized systems, they often serve to overcome coordination problems and result in Pareto-efficient outcomes, for example, but Pareto-efficiency is famously weak, and it has been shown, for example, that different school-choice algorithms are optimal under different circumstances, even using a single ex ante expected social welfare criterion.
In the literature, there is typically a natural or convenient social welfare criterion, but in many real-life contexts, different people have different ideas about the "right" social welfare criterion — which brings us back to voting mechanisms. Insofar as people vote on the basis of ideas at all, people vote primarily on the basis of their conception of the "social good", and only to a much smaller extent, if at all, on "self-interest". One might therefore imagine a two-step procedure in which some mechanism elicits from people their conception of "justice" or "social welfare" in the first step and then asks them for their personal preferences as to their own allocation in the second step, using a mechanism tuned to maximize the criterion selected in the first step.
It is generally the case in theory that a single combined mechanism for doing two things will perform better than multiple separate mechanisms; roughly, if you assume agents are strategic, you sometimes have to "buy off" agents to get them to reveal as much information as possible, and if you combine the mechanisms you can "buy off" the agents in one stage with compensation in the other stage, sometimes at a lower overall cost. There's some level on which it may be useful to think of proportional representation voting schemes themselves in this way; putting aside practical reasons for them related to information-gathering and gaining buy-in from electoral minorities (avoiding e.g. criminal behavior in response to laws perceived to be invalid), one might have a higher-order desire that a committee reflect other people's preferences as well as one's own, even if bills supported by the majority and opposed by the minority are going to be passed under either system, whether by a close vote of proportionally elected representatives or a landslide vote in a chamber dominated by the electoral majority. I suspect there might be other interesting mechanisms that join what are more clearly separate "What is our consensus social goal in terms of heterogeneous and unknown preferences?" and "What are our different preferences, and what outcome therefore maximizes the socially preferred criterion?" questions even in a purely instrumental kind of set-up. One caveat to add before posting this, though, is that I expect the strictly "best" theoretical mechanism in this kind of situation to be weird and complex in some important ways, and thus impractical; it might elucidate more practical conjoined mechanisms, but it might turn out that the best approach in practice is to go back and use a two-stage approach in which agents can readily understand each stage.
 Interestingly, a lot of people know that they vote primarily on the basis of what is "right" rather than their own self-interest, but believe that most other people, especially their opponents, do not!