One of the current challenges in galaxy evolution studies is to establish the mechanisms that govern the escape of ionizing radiation from galaxies. Here, we investigate the connection between Lyman continuum (LyC) escape and the conditions of the circumgalactic medium (CGM), as probed by Lyα halos (LAHs) in emission. We use Lyα and UV continuum imaging data from the Lyα and Continuum Origins Survey (LaCOS), targeting 42 nearby (z≃0.3), star-forming galaxies with LyC observations (escape fractions of ). LaCOS galaxies show extended Lyα emission ubiquitously, with LyC emitters (LCEs) having more compact Lyα morphologies than non-LCEs, and Lyα spatial offsets that do not exceed the extent of the UV continuum. We model the diffuse LAHs using a combined Sérsic plus exponential 2D profile, and find that the characteristic scale length of the Lyα halo is 10 times larger than the UV, on average. We unveil a significant anticorrelation between and the Lyα halo fraction (HF; or contribution of the halo to the total Lyα luminosity), which we propose as a new LyC indicator. Our observations show that halo scale lengths and HFs both scale positively with the optical depth of the neutral gas in the interstellar medium, revealing a picture in which Lyα and LyC photons in LCEs either emerge directly from the central starbursts or escape isotropically, and in the case of Lyα, minimize the number of scattering interactions in a less-extended CGM.