  The $\mu$E4-LEM beamline at Paul Scherrer Institute (PSI, Switzerland) is a
special muon beamline combining the hyprid type surface muon beamline $\mu$E4
with the low energy muon facility (LEM) and delivers ${\mu}^{+}$ with tunable
energy up to 30 keV for low-energy muon spin rotation experiments (LE-$\mu$SR).
We investigate a possible upgrade scenario for the surface muon beamline
$\mu$E4 by replacing the last set of quadrupole triplet with a special solenoid
to obtain 1.4 times original beam intensity on the LEM muon moderator target.
In order to avoid the muon beam intensity loss at the LEM spectrometer due to
the stray magnetic field of the solenoid, three kinds of solenoid models have
been explored and the stray field of the solenoid at the LEM facility is
finally reduced to the magnitude of the geomagnetic field. A more radical
design, "Super-$\mu$E4", has also been investigated for further increasing the
brightness of the low energy muon beam, where we make use of the current
$\mu$E4 channel and all sets of quadrupole triplets are replaced by large
aperture solenoids. Together with the new slanted muon target E, at least 2.9
times the original muon beam intensity can be expected in the Super-$\mu$E4
beamline. Our work demonstrates the feasibility of upgrading surface muon
beamlines by replacing quadrupole magnets with normal-conducting solenoids,
resulting in higher muon rates and smaller beam spot sizes.
