I’m not well-versed in SEO whatsoever, so take this with a grain of salt and do your own research, but:
If you’re the new website you built has the same pages but different links or link structures (www.domain.com/company/about-us vs www.domain.com/about-us), you should use a 301 redirect to indicate to the search engines that the page has permanently moved. This should help to mitigate the issue of the user receiving Error 404.
However, I do think that the best solution, if at all possible, is to match the page links from the old website to the page links on the new website.
If you had an old Services page at www.domain.com/services, and you simply just re-designed the page, I would keep the URL as-is if I could.
That’s not always possible, but if you can do it, it’d be helpful!
This article could be of some assistance to you in mitigating the drop in ranking that comes with re-designing a website: https://www.webfx.com/blog/web-design/fix-avoid-traffic-drops-website-redesign/
Hope this helps.