Context. The measurement of the spin-orbit alignment of hot Jupiters, including a range of orbital and physical properties, can provide information about the evolution of the orbits of this special class of giant planets. Aims. We aim to refine the orbital and physical parameters and determine the sky-projected planet orbital obliquity λ of five eccentric (e ≅ 0.1–0.3) transiting planetary systems: HAT-P-15, HAT-P-17, HAT-P-21, HAT-P-26, and HAT-P-29, whose parent stars have an effective temperature between 5100 K < T_eff < 6200 K. Each of the systems hosts a hot Jupiter, except for HAT-P-26, which hosts a Neptune-mass planet. Methods. We observed transit events of these planets with the HARPS-N spectrograph, obtaining high-precision radial velocity measurements that allow us to measure the Rossiter–McLaughlin effect for each of the target systems. We used these new HARPS-N spectra and archival data, including those from Gaia, to better characterise the stellar atmospheric parameters. The photometric parameters for four of the hot Jupiters were recalculated using 17 new transit light curves, obtained with an array of medium-class telescopes, and data from the TESS space telescope. HATNet time-series photometric data were checked for the signatures of rotation periods of the target stars and their spin axis inclination. Results. From the analysis of the Rossiter-McLaughlin effect, we derived a sky-projected obliquity of λ = 13° ± 6°, λ = −26.3° ± 6.7°, λ = −0.7° ± 12.5°, λ = −26° ± 16°, for HAT-P-15 b, HAT-P-17 b, HAT-P-21 b, and HAT-P-29 b, respectively. Based on theoretical considerations, these small values of λ should be of primordial origin, with the possible exception of HAT-P-21. Due to the quality of the data, we were not able to fully constrain λ for HAT-P-26 b, although a prograde orbit is favoured (λ = 18^° ± 49^°). The stellar activity of HAT-P-21 indicates a rotation period of 15.88 ± 0.02 days, which allowed us to determine its true misalignment angle ψ = 25° ± 16°. Our new analysis of the physical parameters of the five exoplanetary systems returned values compatible with those existing in the literature. Using TESS and the available transit light curves, we reviewed the orbital ephemeris for the five systems and confirmed that the HAT-P-26 system shows transit timing variations, which may tentatively be attributed to the presence of a third body.