: I think you're going backwards. Search engines try to rank pages for different keywords based on the relevance of the content. To do this, they have to determine what the topic of the page

I think you're going backwards. Search engines try to rank pages for different keywords based on the relevance of the content. To do this, they have to determine what the topic of the page is, and what the target audience (based on search keyword) is. So it's hard for a page to rank well on a bunch of completely unrelated keywords unless those are just really rare keywords that few indexed pages contain.

So what is the topic of each page on your site? If you optimize your content to rank well specifically for those topics, and the name of the old company is unrelated to those topics, then your page will naturally drop in ranking for what search engines perceive as an unrelated search term (while simultaneously rising in ranking for the targeted search terms).

Now, if your old company's own site is so poorly optimized that they get beaten in ranking by an unrelated page that simply lists its name in passing, then that's their issue, not yours. They probably have really low quality/spammy/inaccessible content, or are being penalized for spammy link structure. Short of removing their name from indexable content or intentially doing negative SEO on your site so you lose site authority and trust, you really can't do much more about it.

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