The wavelet-based detrending and denoising method TFAW is applied for the first time to EVEREST 2.0-corrected light curves to further improve the photometric precision of almost all K2 observing campaigns (C1-C8, C12-C18). The performance of both methods is evaluated in terms of 6 hr combined differential photometric precision (CDPP), simulated transit detection efficiency, and planet characterization in different SNR regimes. On average, TFAW median 6hr CDPP is $\sim 30\%$ better than the one achieved by EVEREST 2.0 for all observing campaigns. Using the transit least-squares (TLS) algorithm, we show that the transit detection efficiency for simulated Earth-Sun-like systems is ∼8.5 × higher for TFAW-corrected light curves than for EVEREST 2.0 ones. Using the light curves of two confirmed exoplanets, K2-44 b (high-SNR) and K2-298 b (low-SNR), we show that TFAW yields better MCMC posterior distributions, transit parameters compatible with the cataloged ones but with smaller uncertainties and narrows the credibility intervals. We use the combination of TFAW’s improved photometric precision and TLS enhancement of the signal detection efficiency for weak signals to search for new transit candidates in K2 observing campaign 1. We report the discovery of two new K2-C1 Earth-sized planets statistically validated, using the vespa software: EPIC 201170410.02, with a radius of 1.047$^{+0.276}_{-0.257}R_{\oplus }$ planet orbiting an M-type star, and EPIC 201757695.02, with a radius of 0.908$^{+0.059}_{-0.064}R_{\oplus }$ planet orbiting a K-type star. EPIC 201757695.02 is the 9-th smallest planet ever discovered in K2-C1, and the 39-th smallest in all K2 campaigns.