It is a little risky giving recommendations but weathervise the best time to go would probably be July. Add to that the fact you want to beat the crowds, I'd say June is not too late.
Some places I love include the Danish countryside with its people and idyllic villages (stay at eg. Scandic Bygholm Park Horsens, peaceful and a little bit manor like by a park). Why not continue via Odense (H.C. Andersen's home town) to Copenhagen? Maybe along the Swedish coast (nice city Scandic in Halmstad for instance) to Norway and a scenic (short or many days long) cruise of the Fjords. Then you have to decide how north do you want to go? Trondheim? Lapland? Or straight towards Stockholm? After Stockholm maybe the beautiful archipelago between Sweden and Finland (Mariehamn Scandic, difficult to get award nights though - I've not succeeded yet). Why not all the way to Helsinki and from there the train to St.Petersburg (no Scandics) with a side trip to Tallinn (Scandic Palace)? Just some possibilities... As GUWonder said, enjoy your trip!