West to Hiroshima and Fukuoka
The Sanyo Shinkansen continues west from Shin-Osaka to Hiroshima in around ninety minutes on the fastest services, and onward to Hakata (Fukuoka) in a little over two and a half hours — both realistic add-ons to an Osaka-based trip if you have the extra days, and both well within range as a long day trip if pressed, though an overnight is more comfortable.
North to Kanazawa
Kanazawa, on the Sea of Japan side of the country and known for its preserved samurai and geisha districts and the Kenrokuen garden, is generally reached from Osaka via the Thunderbird limited express rather than a direct shinkansen line, with a journey time of a little over two hours.
Building a longer route
Because Osaka sits roughly in the middle of the Tokaido-Sanyo corridor, it works well as a base to branch out in either direction — Tokyo and beyond to the east, Hiroshima and Fukuoka to the west — without the same backtracking a Tokyo-only base would involve for a west-of-Osaka itinerary.