There are quite a few Ryzen 5 4500U laptops just barely with in your price range, offering a significant improvement in performance over the last generation (~20% or more over the 3500U), but many of them are 14". The new Lenovo Ideapad 5 is one of these with a 15.6" display—the 8/128 GB version is $597 but out of stock on Newegg. The 256 GB model is in stock at the moment, but $10 more expensive. If that increase in performance (and power efficiency as well) is worth $150 to you, those are where you want to start looking. In an ordinary office workload though, it wouldn't matter much, and either would be significantly faster than a laptop from 5 years ago. If you give examples of some of the more demanding things you might want to do, things that maybe feel slow on your current laptop, we'd be able to give a firmer answer whether those extra cores will be helpful.
The Vivobook 15 is a good budget laptop, but on Newegg, a similar model with the 3500U and 512 GB SSD is being sold for the same price of $440, refurbished instead of used/like new. It also has Windows 10 Home instead of Pro, though since it's a work laptop you might want to ask if your company has Enterprise volume licensing. I would definitely recommend the Ryzen 5 model if you do decide to go with the last generation Vivobook, though I'll note that if you find a laptop with with a newer Ryzen 3, the 4300U has similar performance to the Ryzen 5 3500U.