  The observed cross sections for $e^+e^-\rightarrow \mu^+\mu^-$ at energies
from 3.8 to 4.6 GeV are measured using data samples taken with the BESIII
detector operated at the BEPCII collider. We measure the muonic widths and
determine the branching fractions of the charmonium states $\psi(4040)$,
$\psi(4160)$, and $\psi(4415)$ decaying to $\mu^+\mu^-$, as well as making a
first determination of the phase of the amplitudes. In addition, we observe
evidence for a structure in the dimuon cross section near 4.220 GeV/$c^2$,
which we denote as $S(4220)$. Analyzing a coherent sum of amplitudes yields
eight solutions, one of which gives a mass of ${M}_{S(4220)}=4216.7 \pm 8.9 \pm
4.1$~MeV/$c^2$, a total width of ${\rm \Gamma^{\rm tot}_{S(4220)}}=47.2 \pm
22.8 \pm 10.5$~MeV, and a muonic width of ${\rm
\Gamma}^{\mu\mu}_{S(4220)}=1.53\pm1.26\pm0.54$~keV, where the first
uncertainties are statistical and the second systematic. The eight solutions
give the central values of the mass, total width, muonic width to be,
respectively, in the range from 4212.8 to 4219.4 MeV/$c^2$, from 36.4 to 49.6
MeV, and from 1.09 to 1.53 keV. The statistical significance of the $S(4220)$
signal is $3.9\sigma$. Correcting the total dimuon cross section for radiative
effects yields a statistical significance for this structure of more than
$7\sigma$.
