The Complete Calibration of the Color-Redshift Relation survey (C3R2) is a spectroscopic program designed to empirically calibrate the galaxy color-redshift relation to the Euclid depth (I-E = 24.5), a key ingredient for the success of Stage IV dark energy projects based on weak lensing cosmology. A spectroscopic calibration sample that is as representative as possible of the galaxies in the Euclid weak lensing sample is being collected, selecting galaxies from a self-organizing map (SOM) representation of the galaxy color space. Here, we present the results of a near-infrared H- and K-band spectroscopic campaign carried out using the LUCI instruments at the LBT. For a total of 251 galaxies, we present new highly reliable redshifts in the 1.3 <= z <= 1.7 and 2 <= z <= 2.7 ranges. The newly-determined redshifts populate 49 SOM cells that previously contained no spectroscopic measurements and almost twice the occupation numbers of an additional 153 SOM cells. A final optical ground-based observational effort is needed to calibrate the missing cells, in particular in the redshift range 1.7 <= z <= 2.7, which lack spectroscopic calibration. In the end, Euclid itself will deliver telluric-free near-IR spectra that can complete the calibration.