Lithium is expected to be fully depleted in red giants due to the dilution they experience during their first dredge-up. However, several Li-rich giants have been observed in the field as well as in open clusters (OCs) with their origin still under discussion. In this work, we explore the Li content of giant stars in 34 different OCs with high-resolution spectroscopy from HARPS (R=110000) by extending our sample from our previous work (Delgado-Mena et al. 2016) in a homogeneous way. A key advantage of OCs is that we can significantly constrain the stellar evolutionary state and thus, the mechanisms associated with the enrichment of Li. We use the spectral synthesis technique to obtain precise stellar parameters and Li abundances for 157 giant stars and found 47 red giants showing clear Li enhancement in 15 open clusters. We infer their evolutionary stage thanks to Gaia photometry and astrometry to fit stellar isochrones with our Bayesian tools (Bossini et al. 2019). We show that our Li-rich stars occupy different regions in the HR diagram. In some cases, they lie near the luminosity bump, where Li is produced by the Cameron-Fowler mechanism. In other cases, these stars are more evolved in their He-burning phase, or approach the asymptotic giant branch. Our sample is part of planet search program in OCs with some giants to have detected substellar companions (Delgado-Mena et al. 2018). Other scenarios we evaluate in this work is the possibility that Li enhancement is triggered by the engulfment of a close-in sub-stellar object which could initiate a process to bring Li in the stellar photosphere.