Jaeger, Linial, Payan, and Tarsi introduced the notion of $A$-connectivity for graphs in 1992, and proved a decomposition for cubic graphs from which $A$-connectivity follows for all 3-edge-connected graphs when $|A|\geq 6$. The concept of $A$-connectivity was generalized to signed graphs by Li, Luo, Ma, and Zhang in 2018 and they proved that all 4-edge-connected flow-admissible signed graphs are $A$-connected when $|A|\geq 4$ and $|A|\neq 5$. We prove that all 3-edge-connected flow-admissible signed graphs are $A$-connected when $|A|\geq 6$ and $|A|\neq 7$. Our proof is based on a decomposition that is a signed-graph analogue of the decomposition found by Jaeger et. al, and which may be of independent interest.