They enter the bigger rivers like the Tay, Tweed, Dee and Spey all year round though the more productive months are generally from late July when the Grilse (1 sea winter fish) arrive and through Sept and Oct when the back end fish come in.
There are only a few decent Spring rivers nowadays like the Helmsdale , N Esk and especially the Dee in Aberdeenshire.
If you go onto a website called Fishpal where you can look at all the noted salmon fisheries and see their catches ;-)