Context. NGC 1566 is a local face-on Seyfert galaxy and is known for exhibiting recurrent outbursts that are accompanied by changes in spectral type. The most recent transient event occurred from 2017 to 2019 and was reported to be accompanied by a change in Seyfert classification from Seyfert 1.8 to Seyfert 1.2. Aims. We aim to study the transient event in detail by analyzing the variations in the optical broad-line profiles. In particular, we intend to determine the structure and kinematics of the broad-line region. Methods. We analyzed data from an optical spectroscopic variability campaign of NGC 1566 taken with the 9.2 m Southern African Large Telescope (SALT) between July 2018 and October 2019 triggered by the detection of hard X-ray emission in June 2018. We supplemented this data set with optical to near-infrared (NIR) spectroscopic archival data taken by VLT/MUSE in September 2015 and October 2017, and investigated the emission from different line species during the event. Results. NGC 1566 exhibits pronounced spectral changes during the transient event. We observe the emergence and fading of a strong power-law-like blue continuum as well as strong variations in the Balmer, He I, and He II lines and the coronal lines [Fe VII], [Fe X], and [Fe XI]. Moreover, we detect broad double-peaked emission line profiles of O Iλ8446 and the Ca IIλλ8498, 8542, 8662 triplet. This is the first time that genuine double-peaked O Iλ8446 and Ca IIλλ8498, 8542, 8662 emission in AGN is reported in the literature. All broad lines show a clear redward asymmetry with respect to their central wavelength and we find indications for a significant blueward drift of the total line profiles during the transient event. The profiles and the FWHM of the Balmer lines remain largely constant during all observations. We show that the double-peaked emission line profiles are well approximated by emission from a low-inclination, relativistic eccentric accretion disk, and that single-peaked profiles can be obtained by broadening due to scale-height-dependent turbulence. Small-scale features in the O I and Ca II lines suggest the presence of inhomogeneities in the broad-line region. Conclusions. We conclude that the broad-line region in NGC 1566 is dominated by the kinematics of a relativistic eccentric accretion disk. The broad-line region can be modeled to be vertically stratified with respect to scale-height turbulence with O I and Ca II being emitted close to the disk in a region with high (column) density, while the Balmer and helium lines are emitted at greater scale height above the disk. The observed blueward drift might be attributed to a low-optical-depth wind launched during the transient event. Except for this wind, the observed kinematics of the broad-line region remain largely unchanged during the transient event.