The hydroxymethyl radical (CH2OH) is one of two structural isomers, together with the methoxy radical (CH3O), that can be produced by abstraction of a hydrogen atom from methanol (CH3OH). In the interstellar medium (ISM), both CH2OH and CH3O are suspected to be intermediate species in many chemical reactions, including those of formation and destruction of methanol. The determination of the CH3O/CH2OH ratio in the ISM would bring important information concerning the formation processes of these species in the gas and solid phases. Interestingly, only CH3O has been detected in the ISM so far, despite the recent first laboratory measurement of the CH2OH rotation-tunneling spectrum. This lack of detection is possibly due to the non-observation in the laboratory of the most intense rotation-tunneling transitions at low temperatures. To support further searches for the hydroxymethyl radical in space, we present a thorough spectroscopic study of its rotation-tunneling spectrum, with a particular focus on transitions involving the lowest quantum numbers of the species. We recorded the rotation-tunneling spectrum of CH2OH at room temperature in the millimeter-wave domain using a frequency multiplication chain spectrometer. A fluorine-induced H-abstraction method from methanol was used to produce the radical. About 180 transitions were observed, including those involving the lowest $N$ and $K_a$ quantum numbers, which are predicted to be intense under cold astrophysical conditions. These transitions were fitted together with available millimeter-wave lines from the literature. A systematic observation of all components of the rotational transitions yields a large improvement of the spectroscopic parameters allowing confident searches of the hydroxymethyl radical in cold to warm environments of the ISM.