"19th century philosophers are not necessary to understand contemporary debates" is largely true because modern debates regurgitate ideas and arguments explored at length since Kant. Here is an example of Searle arguably replaying an argument already discussed by Kant, see What are the problems with the argument for the mind-body dualism from immateriality of thoughts? A text may not be the best way to get into the subject, I start with two short papers instead.
Rosenthal, Theory of Consciousness, a short review (27pp) of the setting of modern debates.
Nagel, What is it like to be a bat?, a short modern classic (9pp). The argument for the impossibility of a physicalist reduction of "what it is like" in a nutshell.
Kim, Philosophy of Mind, a standard introductory text. The author is known for the famous "causal exclusion argument", see Is there a causal influence of the mental on the physical?
Burge, Foundations of Mind, collection of essays on the subject by a prominent modern philosopher, available for free. Some essays are specifically on consciousness, and the last chapter reviews philosophical developments in the area in 1950-2000.
Nagel, The View From Nowhere, the book long version of the argument on the limitations of objectification, and their manifestations in philosophical debates. More subtle and less "pop" than Chalmers, the title has become an idiom.
Carruthers, Phenomenal Consciousness. A Naturalistic Theory An attempt to offer a materialistic theory that takes into account the debates over qualia, in particular the objections such as Nagel's. Is based on the higher-order thought (HOT) theory.
Crick, The Astonishing Hypothesis a 1994 book by a co-discoverer of DNA devoted to his late life passion, neuroscience. Attempts to give a conjectural account of how consciousness emerges using what we know about the brain function, and to set up an empirical programme for exploring it.
Mele, Free: Why Science Hasn’t Disproved Free Will, an up to date philosophical look at the relationship between consciousness and free will in the light of recent neuroscience (Libet experiments, etc.)