This is a request for reference. Where can I find discussion of the Erdős–Rényi random graph in the regime between "giantcomponent" and "fully connected"? e.g. a detailed picture for say, $p_n=\frac{(\log n)^\gamma}{n}$, with $0<\gamma<1$
This may not satisfy your needs, but this recent (Fall 2012) paper, or its references might help:
Among other results, they have a new proof that



"Random Graphs" by Bollobas has an extensive discussion of the thresholds at which various properties appear in the ErdosRenyi model, for example the appearance of the first cycles. It's quite wellwritten, too. I'm familiar with the first edition, which is a little behind the times, but I see a new edition was published in 2001, which should be fairly uptodate (the field has cooled down in the last decade). 

