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Energy market designs with non-merchant storage have been proposed in recent years, with the aim of achieving optimal market integration of storage. In order to handle the time-linking constraints that are introduced in such markets, existing works commonly make simplifying assumptions about the end-of-horizon storage level, e.g., by imposing an exogenous level for the amount of energy to be left for the next time horizon. This work analyzes market properties under such assumptions, as well as in their absence. We find that, although they ensure cost recovery for all market participants, these assumptions generally lead to market inefficiencies. Therefore we consider the design of markets with non-merchant storage without such simplifying assumptions. Using illustrative examples, as well as detailed proofs, we provide conditions under which market prices in subsequent market horizons fail to reflect the value of stored energy. We show that this problem is essential to address in order to preserve market efficiency and cost recovery. Finally, we propose a method for restoring these market properties in a perfect-foresight setting.