A module $M$ is called $\oplus$-$\delta_{ss}$-supplemented if every submodule $X$ of $M$ has a $\delta_{ss}$-supplement $Y$ in $M$ which is a direct summand of $M$ such that $X+Y=M$ and $X\cap Y\leq \Soc_{\delta}(Y)$ where $\Soc_{\delta}(Y)$ is the sum of simple and $\delta$-small submodules of $Y$ and $M=Y\oplus Y'$ for some $Y'\leq M$. Moreover, $M$ is called a completely $\oplus$-$\delta_{ss}$-supplemented module if every direct summand of $M$ is $\oplus$-$\delta_{ss}$-supplemented. Thus, we present two new types of algebraic structures which are stronger than $\delta$-$D_{11}$ and $\delta$-$D_{11}^+$-modules, respectively. In this paper we investigate basic properties, decompositions and ring characterizations of these modules.