  We show how to use Exceptional Field Theory to compute the full Kaluza-Klein
spectra of 10- and 11-dimensional supergravity around deformations of
backgrounds of maximal gauged supergravity by scalar modes that do not form
part of the consistent truncation. This includes deformations of AdS$_4 \times
S^7$ and AdS$_5 \times S^5$ by modes that are not part of the ${\cal N}=8$
supermultiplet. As an application, we compute the full Kaluza-Klein spectrum of
the ${\cal N}=1$ and ${\cal N}=0$ squashed $S^7$. In this example, all
conformal dimensions are captured by a universal formula in terms of the
Casimir operators and additional quantum numbers which organise the spectrum.
