We present the results of a systematic search for molecular hydrogen (H2) in low redshift ($ 0.05 lesssim z lesssim 0.7$) DLAs and sub-DLAs with $N(HI) gtrsim 10^{19.0}$ cm$^{-2}$, in the archival HST/COS spectra. Our core sample is comprised of 27 systems with a median $log N(HI) = 19.6$. On the average, our survey is sensitive down to $log N(H2) = 14.4$ corresponding to a molecular fraction of $log f_{H2} = -4.9$ at the median $N(HI)$. H2 is detected in 10 cases (3/5 DLAs and 7/22 sub-DLAs) down to this $f_{H2}$ limit. The H2 detection rate of $50^{+25}_{-12}$ percent seen in our sample, is a factor of $gtrsim 2$ higher than that of the high-$z$ sample of Noterdaeme et al. (2008), for systems with $N(H2) > 10^{14.4}$ cm$^{-2}$. In spite of having $N(HI)$ values typically lower by a factor of 10, low-$z$ H2 systems show molecular fractions ($log f_{H2}=-1.93pm0.63$) that are comparable to the high-$z$ sample. The rotational excitation temperatures ($T_{01} = 133pm55$ K), as measured in our low-$z$ sample, are typically consistent with high-$z$ measurements. Simple photoionization models favour a radiation field much weaker than the mean Galactic ISM field for a particle density in the range 10 - 100 cm$^{-3}$. The impact parameters of the identified host-galaxy candidates are in the range $10 lesssim rho$ (kpc) $lesssim 80$. We, therefore, conjecture that the low-$z$ H2 bearing gas is not related to star-forming disks but stems from self-shielded, tidally stripped or ejected disk-material in the extended halo.
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