  We construct two new one-parametric families of separated variables for the
classical Lax-integrable Hamiltonian systems governed by a one-parametric
family of non-skew-symmetric, non-dynamical $\mathfrak{gl}(2)\otimes
\mathfrak{gl}(2)$-valued quasi-trigonometric classical $r$-matrices. We show
that for all but one classical $r$-matrices in the considered one-parametric
families the corresponding curves of separation differ from the standard
spectral curve of the initial Lax matrix. The proposed scheme is illustrated by
an example of separation of variables for $N=2$ quasi-trigonometric Gaudin
models in an external magnetic field.
