2nd fermentation using non air tight vessels

[hotrod1203]

Hi I just finished my first fermentation and wanted to know if I could do a second fermentation in a large vessel say a quart jar or something larger with fruit and using a cloth to cover it and leave it out the same as I did for the first fermentation. I realize I would not get any carbonation. If this would be ok to do, would there be any negative affect to leaving the jar out on my counter versus placing it in the refrigerator. My understanding would be that if I placed it in the refrigerator it would just be flavoring but if I leave it on the counter it would actually be fermenting the fruit and flavoring my Kombucha better then just placing it in the fridge. Again, I don’t care about my tea not being fizzy.
Thanks in advance for any help.