  High-quality single crystals are essentially needed for the investigation of
the novel bulk properties of unconventional superconductors. The availability
of such crystals grown by the floating-zone method has helped to unveil the
unconventional superconductivity of the layered perovskite Sr$_2$RuO$_4$, which
is considered as a strong candidate of a topological spin-triplet
superconductor. Yet, recent progress of investigations urges further efforts to
obtain ultimately high-quality crystalline samples. In this paper, we focus on
the method of preparation of feed rods for the floating-zone melting and report
on the improvements of the crystal growth. We present details of the improved
methods used to obtain crystals with superconducting transition temperatures
$T_\mathrm{c}$ that are consistently as high as 1.4 K, as well as the
properties of these crystals.
