Technically you could copy/paste the contents of "Small on Top [GAME VERSION]/Small on Top/www/save" into the new one. Unfortunately though, RPGMaker is a fickle thing that doesn't like it when newer scripts are added to older save files. The reason each update comes with some prepackaged save files is because one can never know if/what kinda glitches or bugs may occur if you use save files from a previous version. It might be completely fine, or certain new things won't even appear in the game at all. And with the public version, there's an entire year's worth of Patreon updates inbetween. Here's an example when I first started playing: When I downloaded a newer version, the Fire Dungeon got it's underwater section, but somehow I wasn't able to enter the water. Turned out it was because I was using an older save file, that didn't have that section yet. Don't worry though, the preset save files put you either at the pirate ship before you leave the Western Island, or just before you leave Pine Island. All the files come with decently leveled characters, you got a lot of gold, and each item available in the game's in your inventory at least once, plus in 0.16 you can even use the ID machine in the Gloomy Forest to change the Adventurer's name to what you were using before. And all the things/quests that can be done/unlocked in the game up to this point has already been done in the save files, so you shouldn't feel like you're losing any progress if you use them.
But you MUST make a copy of the save file folder before you delete the old version, because unlike with engines like Ren'py, which store the save files in a separate folder in your documents, RPGMaker keeps it's save files inside the actual game folder. So, deleting the game will also mean you lose the save folder.