Cycle double covers and non-separating cycles

Which $2$-regular subgraph $R$ of a cubic graph $G$ can be extended to a cycle double cover of $G$? We provide a condition which ensures that every $R$ satisfying this condition is part of a cycle double cover of $G$. As one consequence, we prove that every $2$-connected cubic graph which has a decomposition into a spanning tree and a $2$-regular subgraph $C$ consisting of $k$ circuits with $k\leq 3$, has a cycle double cover containing $C$.