: Upgrade your motherboard BIOS, purchase and install the Ryzen 5 5600 ($137) and sell the Ryzen 5 1600 used on Ebay, Facebook marketplace, etc ($50-$60).
: Please contact Gigabyte support to confirm that the Gigabyte GA A320M S2H supports the Ryzen 5 5600 (non-X). And once you do, then please follow the upgrade notes on Gigabyte's BIOS page to the letter. Also, be aware that the Gigabyte GA A320M S2H has three revision numbers.
and happy holidays. If I were in your position, I would be pondering the seven options below
1. keep the Ryzen 5 1600
2. upgrade to Ryzen 5 5600 ($137 USD on Amazon)
3 upgrade to the Ryzen 7 5700X ($197 USD on Amazon)
4. upgrade to Ryzen 7 5800X3D ($336 USD on Amazon)
5. upgrade to AM5 platform Ryzen 5 7600X ($240 USD on Amazon), and buy AM5 socket motherboard and DDR5 RAM ($$$)
6. upgrade to future AM5 platform Ryzen 5 7600 (non-X), which is anticipated to be announced at CES January 5th 2023, for less than the 'X' version.
7. upgrade to future AM5 platform Ryzen 7 7800X3D, which is anticipated at some point in the future.
: Since you're already on the existing AM4 platform, and your running a mid-tier performance RTX 3060 Ti which wouldn't noticeably benefit from R7 5800X3D, my #1 upgrade recommendation
would be to buy a Ryzen 5 5600 (non-X). My #2 upgrade would be to the Ryzen 7 5800X3D, but I would only consider it if upgrading to no less than a RTX 3080, and once the R7 5800X3D price has dropped below $300 (ideally < $250). As for the Ryzen 7 5800X that you were considering, consider putting that saved $100 toward my suggestion immediately below.
Might as well
: One last thing... I noticed that you have a 256GB SSD and two 1TB HDD's. Since your motherboard has a single M.2 slot, I strongly encourage you to consider purchasing this 2TB SSD, and getting rid the two hard drives. That motherboard doesn't support PCIe 4 speeds, but it would still be a monstrous upgrade for your build. The listing that I pasted below is on NewEgg, because their price for that drive is noticeably cheaper than Amazon's. Solidigm is formely Intel's SSD division, and I've personally purchased several of these drives. They are excellent, especially for the price.
Solidigm P41 Plus Series 2TB
AMD Ryzen 5 5600