  We study the fixed point that controls the IR dynamics of QED in $d = 4 -
2\epsilon$. We derive the scaling dimensions of four-fermion and bilinear
operators beyond leading order in $\epsilon$-expansion. For the four-fermion
operators, this requires the computation of a two-loop mixing that was not
known before. We then extrapolate these scaling dimensions to $d = 3$ to
estimate their value at the IR fixed point of QED$_3$ as function of the number
of fermions $N_f$. The next-to-leading order result for the four-fermion
operators corrects significantly the leading one. Our best estimate at this
order indicates that they do not cross marginality for any value of $N_f$,
which would imply that they cannot trigger a departure from the conformal
phase. For the scaling dimensions of bilinear operators, we observe better
convergence as we increase the order. In particular, $\epsilon$-expansion
provides a convincing estimate for the dimension of the flavor-singlet scalar
in the full range of $N_f$.
