  Background: Whether or not the $\alpha$ ($^4$He nucleus) clustering exists in
the medium-mass region of nuclear systems is a fundamental and intriguing
question. However, the recent analysis of the $\alpha$ knockout reaction on
$^{48}$Ti [Phys. Rev. C 103, L031305 (2021)] poses a puzzle: The microscopic
wave function gives an $\alpha$ knockout cross section that is two orders of
magnitude smaller than the experiment, while basic nuclear properties such as
the charge radius and the electromagnetic transition probabilities are well
explained.
  Purpose: The ground-state structure of $^{48}$Ti is investigated by using
proton- and $\alpha$-nucleus elastic scattering at a few to several hundred
MeV, which offers different sensitivity to the region of the nuclear density
profiles.
  Method: Four types of density distributions, the $jj$-coupling shell model
and three cluster model configurations, are generated in a single scheme by the
antisymmetrized quasi-cluster model (AQCM). The angular distribution of the
proton- and $\alpha$-$^{48}$Ti elastic scattering cross sections are obtained
with a reliable high-energy reaction theory, the Glauber model.
  Results: The $jj$-coupling shell model configuration is found to best
reproduce the proton-nucleus elastic scattering cross section. On the other
hand, the trace of the $\alpha$ cluster structure in the tail region of the
wave function is embedded in the $\alpha$-nucleus elastic scattering cross
section.
  Conclusion: Our results suggest that the structure of the nucleus changes as
a function of distance from the center, from the $jj$-coupling shell model
structure in the surface region to the $\alpha$+$^{44}$Ca cluster structure in
the tail region. This picture is consistent with the finding of the $\alpha$
knockout reaction on $^{48}$Ti.
