A generalization of the ensemble of non-terminated systematic LDPC convolutional codes developed in our previous work is proposed that allows us to design codes with lower rates than the original structure. We show that over the BEC, the modified codes have improved asymptotic and finite-length behavior and we determine the operational anytime exponent. Having shown the advantages of lowering the rate of the code, we propose a feedback protocol that permits encoder and decoder to operate at a variable rate. The rate is set on-the-fly and depends on the decoding success of the decoder. We describe the construction of the variable rate code structure and demonstrate by simulations the superiority of the variable rate scheme as compared to a scheme using a fixed rate.