Simplicial sets are presheaves on the simplex category $\Delta$, while augmented simplicial sets are presheaves on $\Delta_+$, the augmented simplex category. Because Day convolution allows us to lift monoidal structures on a category $\mathcal{C}$ to its category of presheaves $\mathrm{Sets}^{\Delta^\circ}$, it is therefore of interest to find monoidal structures on $\Delta$ and $\Delta_+$, as these then provide "natural" monoidal structures on simplicial sets.
The only monoidal structure I know of is the ordinal sum of $\Delta_+$ (which is not braided), whose Day convolution gives the join of simplicial sets, and whose internal hom is given by $$[X,Y]_n=\mathrm{hom}_{\mathrm{Sets}^{\Delta^\circ_+}}(X,\mathrm{Dec}^{n+1}Y)$$
Is this the only monoidal structure on $\Delta_+$? If not, what other monoidal structures are there on $\Delta_+$, and what are there on $\Delta$?
This question has been answered on MathOverflow.