In this paper, we propose a spectral-efficient LoRa (SE-LoRa) modulation scheme with a low complexity successive interference cancellation (SIC)-based detector. The proposed communication scheme significantly improves the spectral efficiency of LoRa modulation, while achieving an acceptable error performance compared to conventional LoRa modulation, especially in higher spreading factor (SF) settings. We derive the joint maximum likelihood (ML) detection rule for the SE-LoRa transmission scheme that turns out to be of high computational complexity. To overcome this issue, and by exploiting the frequency-domain characteristics of the dechirped SE-LoRa signal, we propose a low complexity SIC-based detector with a computation complexity at the order of conventional LoRa detection. By computer simulations, we show that the proposed SE-LoRa with low complexity SIC-based detector can improve the spectral efficiency of LoRa modulation up to $445.45\%$, $1011.11\%$, and $1071.88\%$ for SF values of $7$, $9$, and $11$, respectively, while maintaining the error performance within less than $3$ dB of conventional LoRa at symbol error rate (SER) of $10^{-3}$ in Rician channel conditions.