Depends what you want to do.
If you want to catch a highland games it would need to be September.
Pitlochry Games 11th September, Peebles Games in the Scottish Borders 12th September, and Invercharron Games, north of Inverness on 18th September and then that's the season over.
Be sure to visit the rolling hills of the Scottish Southern Uplands and the beautiful Solway coastline, south of the Glasgow/Edinburgh central belt,