Aims. The transition between atomic and molecular hydrogen is associated with important changes in the structure of interstellar clouds, and marks the beginning of interstellar chemistry. Most molecular ions are rapidly formed (in ion–molecule reactions) and destroyed (by dissociative recombination) in the diffuse ISM. Because of the relatively simple networks controlling their abundances, molecular ions are usually good probes of the underlying physical conditions including, for instance the fraction of gas in molecular form or the fractional ionization. In this paper we focus on three possible probes of the molecular hydrogen column density, HCO⁺, HOC⁺, and CF⁺. Methods. We presented high-sensitivity ALMA absorption data toward a sample of compact H II regions and bright QSOs with prominent foreground absorption, in the ground-state transitions of the molecular ions HCO⁺, HOC⁺, and CF⁺ and the neutral species HCN and HNC, and from the excited-state transitions of C₃H⁺(4-3) and ¹³CS(2-1). These data are compared with Herschel absorption spectra of the ground-state transition of HF and p-H₂O. Results. We show that the HCO⁺, HOC⁺, and CF⁺ column densities are well correlated with each other. HCO⁺ and HOC⁺ are tightly correlated with p-H₂O, while they exhibit a different correlation pattern with HF depending on whether the absorbing matter is located in the Galactic disk or in the central molecular zone. We report new detections of C₃H⁺ confirming that this ion is ubiquitous in the diffuse matter, with an abundance relative to H₂ of ~7 × 10⁻¹¹. Conclusions. We confirm that the CF⁺ abundance is lower than predicted by simple chemical models and propose that the rate of the main formation reaction is lower by a factor of about 3 than usually assumed. In the absence of CH or HF data, we recommend to use the ground-state transitions of HCO⁺, CCH, and HOC⁺ to trace diffuse molecular hydrogen, with mean abundances relative to H₂ of 3 × 10⁻⁹, 4 × 10⁻⁸, and 4 × 10⁻¹¹, respectively, leading to sensitivity N(H₂)/ ∫ τdv of 4 × 10²⁰, 1.5 × 10²¹, and 6 × 10²² cm⁻² /km s⁻¹, respectively.