  Detailed knowledge of the variability of classical Cepheids, in particular
their modulations and mode composition, provides crucial insight into stellar
structure and pulsation. However, tiny modulations of the dominant radial-mode
pulsation were recently found to be very frequent, possibly ubiquitous in
Cepheids, which makes secondary modes difficult to detect and analyse, since
these modulations can easily mask the potentially weak secondary modes. The aim
of this study is to re-investigate the secondary mode content in the sample of
OGLE-III and -IV single-mode classical Cepheids using kernel regression with
adaptive kernel width for pre-whitening, instead of using a constant-parameter
model. This leads to a more precise removal of the modulated dominant
pulsation, and enables a more complete survey of secondary modes with
frequencies outside a narrow range around the primary. Our analysis reveals
that significant secondary modes occur more frequently among first overtone
Cepheids than previously thought. The mode composition appears significantly
different in the Large and Small Magellanic Clouds, suggesting a possible
dependence on chemical composition. In addition to the formerly identified
non-radial mode at $P_2 \approx 0.6\ldots 0.65 P_1$ (0.62-mode), and a cluster
of modes with near-primary frequency, we find two more candidate non-radial
modes. One is a numerous group of secondary modes with $P_2 \approx 1.25 P_1$,
which may represent the fundamental of the 0.62-mode, supposed to be the first
harmonic of an $l \in \{7,8, 9\}$ non-radial mode. The other new mode is at
$P_2 \approx 1.46 P_1$, possibly analogous to a similar, rare mode recently
discovered among first overtone RR Lyrae stars.
