Will start picking them up in October but the fish will generally hang around until June. Best time is probably after the closure to about mid April, although this will depend on where abouts youre heading.
Arno Bay and Whyalla are obviously awesome, but you dont need to head that far afield. Plenty of big fish are landed off Adelaide, with a couple of charters operating out of Glenelg that seem to get good catches. As Stash mentioned above, you can head south to Cape Jervois/KI/Wirrina, or north to Ardrossen and the Yorke Penninsula. All within 1.5hrs drive of the city.
Good luck.