  We report on observations of the active K2 dwarf $\epsilon$ Eridani based on
contemporaneous SPIRou, NARVAL, and TESS data obtained over two months in late
2018, when the activity of the star was reported to be in a non-cyclic phase.
We first recover the fundamental parameters of the target from both visible and
nIR spectral fitting. The large-scale magnetic field is investigated from
polarimetric data. From unpolarized spectra, we estimate the total magnetic
flux through Zeeman broadening of magnetically sensitive nIR lines and the
chromospheric emission using the CaII H & K lines. The TESS photometric
monitoring is modeled with pseudo-periodic Gaussian Process Regression.
Fundamental parameters of $\epsilon$ Eridani derived from visible and
near-infrared wavelengths provide us with consistent results, also in agreement
with published values. We report a progressive increase of macroturbulence
towards larger nIR wavelengths. Zeeman broadening of individual lines
highlights an unsigned surface magnetic field $B_{\rm mono} = 1.90 \pm 0.13$
kG, with a filling factor $f = 12.5 \pm 1.7$% (unsigned magnetic flux $Bf = 237
\pm 36$ G). The large-scale magnetic field geometry, chromospheric emission,
and broadband photometry display clear signs of non-rotational evolution over
the course of data collection. Characteristic decay times deduced from the
light curve and longitudinal field measurements fall in the range 30-40 d,
while the characteristic timescale of surface differential rotation, as derived
through the evolution of the magnetic geometry, is equal to $57 \pm 5$ d. The
large-scale magnetic field exhibits a combination of properties not observed
previously for $\epsilon$ Eridani, with a surface field among the weakest
previously reported, but also mostly axisymmetric, and dominated by a toroidal
component.
