Context. Type Ia Supernovae (SNe Ia) have been used as standardizable candles in the optical wavelengths to measure distances with an accuracy of similar to 7% out to redshift z similar to 1 : 5. There is evidence that in the near-infrared (NIR) wavelengths SNe Ia are even better standard candles, however, NIR observations are much more time-consuming. Aims. We aim to test whether the NIR peak magnitudes could be accurately estimated with only a single observation obtained close to maximum light, provided that the time of B band maximum, the B - V color at maximum and the optical stretch parameter are known. Methods. We present multi-epoch UBVRI and single-epoch J and H photometric observations of 16 SNe Ia in the redshift range z = 0 : 037 0 : 183, doubling the leverage of the current SN Ia NIR Hubble diagram and the number of SNe beyond redshift 0.04. This sample was analyzed together with 102 NIR and 458 optical light curves (LCs) of normal SNe Ia from the literature. Results. The analysis of 45 NIR LCs with well-sampled first maximum shows that a single template accurately describes the LCs if its time axis is stretched with the optical stretch parameter. This allows us to estimate the peak NIR magnitudes of SNe with only few observations obtained within ten days from B-band maximum. The NIR Hubble residuals show weak correlation with Delta M-15 and the color excess E(B V), and for the first time we report a potential dependence on the J(max) - H-max color. With these corrections, the intrinsic NIR luminosity scatter of SNe Ia is estimated to be similar to 0.10 mag, which is smaller than what can be derived for a similarly heterogeneous sample at optical wavelengths. Analysis of both NIR and optical data shows that the dust extinction in the host galaxies corresponds to a low R-V similar or equal to 1.8-1.9. Conclusions. We conclude that SNe Ia are at least as good standard candles in the NIR as in the optical and are potentially less affected by systematic uncertainties. We extended the NIR SN Ia Hubble diagram to its nonlinear part at z similar to 0 : 2 and confirmed that it is feasible to accomplish this result with very modest sampling of the NIR LCs, if complemented by well-sampled optical LCs. With future facilities it will be possible to extend the NIR Hubble diagram beyond redshift z similar or equal to 1; and our results suggest that the most efficient way to achieve this would be to obtain a single observation close to the NIR maximum.