High precision macroscopic quantum control in interacting light-matter systems remains a significant goal toward novel information processing and ultra-precise metrology. We show that the out-of-equilibrium behavior of a paradigmatic light-matter system (Dicke model) reveals two successive stages of enhanced quantum correlations beyond the traditional schemes of near-adiabatic and sudden quenches. The first stage features magnification of matter-only and light-only entanglement and squeezing due to effective nonlinear self-interactions. The second stage results from a highly entangled light-matter state, with enhanced superradiance and signatures of chaotic and highly quantum states. We show that these new effects scale up consistently with matter system size, and are reliable even in dissipative environments.Many-body quantum dynamics are at the core of many natural and technological phenomena, from understanding of superconductivity or magnetism, to applications in quantum information processing as in adiabatic quantum computing [1]. Critical phenomena, defect formation, symmetry breaking, finite-size scaling are all aspects that emerge from the collective properties of the system [2]. Spin networks, many-body systems composed of the simplest quantum unit, are an obvious starting point to understand those phenomena, as they enclose much of their complex behavior in a highly controllable and tractable way. However, if the system under investigation includes a radiation subsystem, new opportunities arise for monitoring and characterizing the resulting collective phenomena [3,4]. By devising driving protocols of the light-matter interaction, high precision macroscopic control then becomes a possibility, regardless of whether the focus is on the matter subsystem, the light, or the composite manipulation of both. This is particularly true for the Dicke model (DM) [5], which is the subject of the present work.The DM describes a radiation-matter system which, despite its simplicity, exhibits a wide arrange of complex collective phenomena, many of them specifically associated with the existence of a quantum phase transition (QPT) [6]. Experimental realizations of the DM have been presented in different settings, from proposed realizations in circuit quantum electrodynamics [7], to the recent very successful demonstrations of DM superradiance in various cold atom experiments [8]. While the light and matter properties in the equilibrium ground state are relatively well known [9][10][11][12][13], its fully quantum out-of-equilibrium critical behavior is just starting to be understood [14,15].In the DM, both the matter and field are known to act as mediators of an effective nonlinear self-interaction involving each other [10,13]. These nonlinear interactions produce interesting phenomena in both atomic and optical systems [16,17]. Among the most relevant effects, there is the strong collapse and revival of squeezing [18,19], which in many matter states can be related to atom-atom entanglement [20]. Applications of such eff...