The new firmware is now available update for Samsung Galaxy S 3 Mini. This update not come with any new features and only improve the phone system performance. This firmware version is I8190XXAMG4 and only for Galaxy S 3 Mini I8190.
You can go to setting >software update to check for update your phone to this latest firmware. If your phone is running custom software or running original rooted firmware then you cant check the firmware update on your phone. You need to download the stock firmware and manually flash into your phone via odin. Please make sure that your phone model is compatible to this firmware!
Download Android 4.1.2 I8190XXAMG4 Firmware for Galaxy S 3 Mini: Link
Note: Manual flash Android firmware to Galaxy S 3 Mini via Odin will void the phone warranty. So, take it on your own risk.