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We present further results from the MUSE-ALMA Haloes survey, which includes 79 galaxies associated with strong H I absorption at z ∼ 0.5. As part of this effort, our ALMA Cycle 10 Large Program contributed new observations of 39 systems. This expands on the initial set of 21 systems in the MUSE-ALMA Haloes survey, bringing the total to 60 galaxies. Among the newly observed systems, we detect CO line emission in nine galaxies, corresponding to a 23% detection rate in a sample not selected by metallicity. When combined with prior MUSE-ALMA Haloes data, our total CO detection count rises to 12 out of 60 galaxies (20%), which effectively doubles the number of detected CO-emitting H I-selected galaxies at z ∼ 0.5 and probes a factor of ∼1.2 dex deeper in M_H2 than earlier absorber studies. These sources, which were selected based on known circumgalactic H I gas, span a wide range of stellar masses and metallicities, providing a unique view of gas-rich environments. By comparing the molecular gas properties, traced through CO(2-1) and CO(3-2) transitions with existing information of their physical properties, such as star formation rates (SFRs) and gas-phase metallicities from VLT/MUSE and HST spectroscopy, we investigate how these systems relate to the population of normal star-forming galaxies at similar redshift. Our deep, unbiased CO observations of H I-selected galaxies reveal a dual behaviour in star formation efficiency. Low-M_H2 systems form stars efficiently and follow the scaling relations of main-sequence galaxies, while high-M_H2 systems exhibit suppressed star formation and lower-than-expected stellar masses, likely reflecting ongoing gas accretion or environmental regulation. This diversity indicates that H I absorbers trace both evolved, actively star-forming galaxies and younger or dynamically influenced systems that are still building their gas reservoirs. By reaching molecular gas masses more than 1 dex below previous studies, our survey provides a key step towards completing the baryon census at z ∼ 0.5 and characterising the molecular phase of the broader H I-selected population.