  We carefully examine critical metrology and present an improved critical
quantum metrology protocol which relies on quenching a system exhibiting a
superradiant quantum phase transition beyond its critical point. We show that
this approach can lead to an exponential increase of the quantum Fisher
information in time with respect to existing critical quantum metrology
protocols relying on quenching close to the critical point and observing power
law behaviour. We demonstrate that the Cram\'er-Rao bound can be saturated in
our protocol through the standard homodyne detection scheme. We explicitly show
its advantage using the archetypal setting of the Dicke model and explore a
quantum gas coupled to a single-mode cavity field as a potential platform. In
this case an additional exponential enhancement of the quantum Fisher
information can in practice be observed with the number of atoms $N$ in the
cavity, even in the absence of $N$-body coupling terms.
