As I'm lucky enough to have access to that article, I'm going to extract whatever I can find to answer your question.
To begin with, innate immunity must have evolved first - we can see it at almost all stages of evolution. According to Cooper & Herrin, ever since aerobic respiration gave rise to multicellular organisms which in turn needed protection from invasion by single-cell organisms.
They state that around 500 million years ago, the first adaptive immune systems evolved in vertebrates, but do not explain how although they attempt to. Instead, they explain why we are not able to discern at the moment how this evolution came about. The main reason given is that it is unknown when some key cells evolved (namely natural killer cells and dendritic cells) Additionally, mice and humans evolved two different kinds of natural killer cell receptors relatively recently while sharing a common ancestor only quite a long time ago.
Apparently jawed and jawless vertebrates have also evolved two different kinds of adaptive immune systems. They both seem to rely on the same mechanisms but on a different molecular and genetic basis. Cooper & Herrin conclude that at the current level of research, we are not able to determine the evolution of our immune system.
Source: How did our complex immune system evolve?
Personally, if I'm allowed to give in to the temptation of speculation a bit, I would go by a similar strategy as the evolution of Krebs cycle - look which bits make sense even without the rest. There is of course the major obstacle that the cells of the immune system require an entirely new branch in haemopoiesis: all innate immune cells (aside from natural killer, NK cells) derive from the common myeloid progenitor, whereas all adaptive immune cells (and NK cells) derive from the common lymphoid progenitor.
I think this may even be a first hint: NK cells are innate immune cells but they derive from a different lineage at the highest level - they may have evolved as the first lymphoid cells after the myeloid innate immune cells. The next closest related cell is the cytotoxic T cell (CD8+, aka Tc), which utilises the exact same killing mechanisms (perforin+granzymes and Fas ligand) and attaches to other cells in almost the same way as the NK cell (immune synapse). It is also independent and can kill on its own, as long as it recognises its specific antigen. This is the main difference and also a very large step - the generation of the specific T cell receptor. Once the specific T cells receptor is developed, the step to helper T cells (CD4+, Th) is not far, although these perform a relatively bystander-like role in the immune response, steering the other immune cells from behind the scenes and not directly effecting any killing of the pathogen. Therefore I would guess that before helper T cells, B cells evolved, which are capable of producing antibodies. Antibodies themselves are very similar to the T cell receptor (not in gross structure but in the underlying genetic domains and their processing), so a 'detachable' antigen-specific T cell receptor may have evolved and ultimately caused the development of a new cell type. Once all these components are present, it is necessary (or beneficial) to control which cell types are the most active and in which way they act at the site of infection. This is exactly what helper T cells do, by steering the local immune response either into a Th1 (cytotoxic) or a Th2 ('humoral', anti-parasitic) response direction.