  The precision of recent experiments such as ${\it Planck}$ have allowed us to
constrain standard and non-standard physics (e.g., due to dark matter
annihilation or varying fundamental constants) during the recombination epoch.
However, we can also probe this era of cosmic history using model-independent
variations of the free electron fraction, $X_{\rm e}$, which in turn affects
the temperature and polarization anisotropies of the cosmic microwave
background. In this paper, we improve on the previous efforts to construct and
constrain these generalised perturbations in the ionization history, deriving
new optimized eigenmodes based on the full Planck 2015 likelihood data,
introducing the new module 'FEARec++'. We develop a direct likelihood sampling
method for attaining the numerical derivatives of the standard and non-standard
parameters, and discuss complications arising from the stability of the
likelihood code. We improve the amplitude constraints of the ${\it Planck}$
2015 principal components constructed here, $\mu_1=-0.09\pm0.12$,
$\mu_2=-0.17\pm0.20$ and $\mu_3=-0.30\pm0.35$, finding no indication for
departures from the standard recombination scenario. The third mode error is
reduced by $60\%$. We utilise an efficient eigen-analyser that keeps the
cross-correlations of the first three eigenmodes to $\xi_{\mu,\mu'}<0.1\%$
after marginalisation for all the considered data combinations. We also propose
a new projection method for estimating constraints on the parameters of
non-standard recombination. Using our eigenmodes, this allows us to recreate
the ${\it Planck}$ constraint on the two-photon decay rate, $A_{\rm
2s1s}=7.60\pm0.64$, giving an estimate to within $\simeq 0.05\sigma$ of the
full MCMC result. The improvements on the analysis using the ${\it Planck}$
data will allow us to implement this new method for analysis with fundamental
constant variations in the future.
