In this paper, we propose a floating-gate-based synaptic transistor with two independent control gates that implement both offline and online learning. Unlike previous research on double-gated synaptic transistors, the proposed device is capable of online learning without facing a fan-out problem. Basic operation of the device was verified and a program/erase scheme based on Fowler-Northeim tunneling is suggested for the multi-conductance utilization of the synaptic device. With the proposed P/E scheme, an offline-trained single-layered hardware-based spiking neural network was simulated for MNIST classification, resulting in 87.37% classification accuracy with 10% conductance variation. To alleviate this performance degradation, the online learning method is applied on the offline-trained SNN by reusing 3,000 training images. The effectiveness of the proposed method is also verified under existence of the synaptic weight variance. As a result, up to 86.89% of the performance degradation is alleviated.