All fluorescence super-resolution microscopy techniques present trade-offs between, for example, resolution, acquisition speed, and live-cell compatibility. Structured illumination microscopy (SIM) improves the resolution through successive imaging of the sample under patterned illumination. SIM can be fast and typically uses low light levels well suited for live cell imaging. However, in its linear form, the resolution gain of SIM is limited by the pattern frequency to a 2-fold improvement over the diffraction limit. Super-resolution optical fluctuation imaging (SOFI) is another low-light level method that achieves higher resolution through the computation of spatiotemporal cross-cumulants of a time series of stochastically blinking fluorescent emitters. The resolution is theoretically enhanced by a factor n, where n is the cumulant order. In practice, it is restricted to smaller orders due to limited signal-to-noise and the need for many frames for good statistics. Here, we demonstrate the experimental combination of SOFT with SIM, where we use SOFI as a source of nonlinearity to further enhance the SIM resolution. We present two implementations of SIM combined with self-blinking dyes for SOFT. We first introduce a new Michelson SIM setup for achromatic high-efficiency (40%) illumination and fast structured pattern projection. We use the setup to acquire single- and two-color SIM data of blinking emitters with up to 2.4-fold image resolution increase and discuss the SOFI-SIM reconstruction challenges. We applied the same concept to realize SOFI-SIM on a flat-fielded, high-throughput instant SIM (iSIM) setup, achieving similar resolution enhancement and demonstrating the versatility of our approach. We established an experimental proof-of-principle of a wide-field combination of SOFT with SIM and iSIM for large fields-of-view, improving SIM resolution without increased complexity of the setup.