No matter if it's a full season or not, Split vicinity has so much to offer, it just invites you to explore it on some day trip. Naturally, during the summer choices are easy; enjoying sea is just too tempting. However, even in this part of the year Dalmatia's climate makes easily possible any attempt to explore natural and historical heritage which is hidden from the islands to the inland Dalmatia.
Options are literally endless. You can be a mountain lover, and start your ascend to Kozjak, Mosor or Biokovo. All those hiking spots have excellently marked paths, spectacular views toward sea and hinterland, and it's also possible to spend a night in mountain homes.
If you prefer cycling, I'm sure you can find more than enough routes, for example on this web maintained by real biking enthusiasts.
You can also tour cultural landmarks and historical monuments, or enjoy in inland Dalmatia's gastronomy which is equally great like the one by the sea. To cut a long story short, whatever you can think of. To make things even better, all this is not just a part of tourist offer, since you will meet a plenty of locals enjoying the same.
Actually, it's the hardest to decide where and how to go. Good starting point might be the Central Dalmatia Tourist Board's web site, with just enough options for day trips.
It's always possible to hire an agency as a planning assistant. Personally, I prefer self-exploring. Important tool in finding wandering destinations is a small, but endlessly useful booklet titled "Excursions from Split-the Split Riviera and Hinterland"; you can read about it here. It's available in any better bookstore, in Croatian, English and German. This booklet is really a "bible" of excursions around Split, with plenty of destinations reachable with local buses. Every single location is described in details, with tips what to see, how to reach it, where to walk, etc.
Whichever option you choose, I'm sure you'll enjoy it. And don't think you will run out of potential destinations. You can live in Dalmatia all your life, and still not to see a small part of everything it has to offer.