While the power supply probably isn't great (they don't even seem to have listed it in the specifications there), if it's a 450 watt unit and has at least one 8-pin PCIe power cable, I suspect it could likely handle a GTX 1660 or 1660 Ti. Those cards are fairly efficient, and should only draw around 120-130 watts under load, which should only be around 50 watts or so more than your current card. Nvidia's specifications state that a 450 watt PSU should be fine for those cards, and even with one of them installed, all of your system's components together likely wouldn't be drawing much more than 300 watts from the PSU under load.
A higher-quality PSU might be more reliable, and could reduce the chances of damage happening to your components in the event of a power supply failure, but if your existing unit has the necessary 8-pin power cable, it would most likely be capable of running one of these cards.
As for AMD's current offerings around the $200-$300 price range, they are more power-hungry, and you would probably want a PSU upgrade for one of those. It seems likely that they will launch new, more efficient cards in that range sometime in the coming months, but for now, the only upcoming cards they have announced are set to be priced $380 or more when they launch next week.