  We present an improved method for the precise reconstruction of cosmic ray
air showers above $10^{17}$ eV with sparse radio arrays. The method is based on
the comparison of predictions for radio pulse shapes by CoREAS simulations to
measured pulses. We applied our method to the data of Tunka-Rex, a 1 km$^2$
radio array in Siberia operating in the frequency band of 30-80 MHz. Tunka-Rex
is triggered by the air-Cherenkov detector Tunka-133 and by scintillators
(Tunka-Grande). The instrument collects air-shower data since 2012. The present
paper describes updated data and signal analyses of Tunka-Rex and details of a
new method applied. After efficiency cuts, when Tunka-Rex reaches its full
efficiency, the energy resolution of about 10% given by the new method has
reached the limit of systematic uncertainties due to the calibration
uncertainty and shower-to-shower fluctuations. At the same time the shower
maximum reconstruction is significantly improved up to an accuracy of 35
g/cm$^2$ compared to the previous method based on the slope of the lateral
distribution. We also define and now achieved conditions of the measurements,
at which the shower maximum resolution of Tunka-Rex reaches a value of 25
g/cm$^2$ and becomes competitive to optical detectors. To check and validate
our reconstruction and efficiency cuts we compare individual events to the
reconstruction of Tunka-133. Furthermore, we compare the mean of shower maximum
as a function of primary energy to the measurements of other experiments.
