Hi Alan,
First, a few things to consider:
- Did your site actually get hit with a penalty? You talk about removing unnatural links. Was that just to be safe, or something else?
- If your 1st site revision changed site urls and generated a lot of 404s, without proper redirection, you'd get the exact result you're describing.
- Did you get new copy written for your 1st site revision? If so, and you didn't write it, take a few sentences from different pages and search for them, in quotes, on Google. Make sure they weren't plagiarized, because that would explain a drop in traffic, too.
All of that said, here's what I'd do:
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**Definitely **work to improve engagement. Engagement is good for your business anyway, so even if it had zero effect on visits, you'd make existing visitors happier. And, Google at least rewards you because of the secondary effects of a great UX: More attention and citation, more positive reviews, fewer bounce backs, etc. And yes, there's some evidence Google rewards great UX, whether deliberately or as a side effect.
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Look for lost links and repair them. Use OpenSiteExplorer and get the Top Pages report. Look for all pages that respond with a 404. Put those pages back (if you had that page before), build a page at that location (if you never had that page) or do a 301 redirect from the page URL to a 'real' page (the easiest fix).
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Check site performance. Did site load speed take a huge hit with the new design?
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Look at your log files. Go all the way back to before the 1st site relaunch. Compare Googlebot activity on your site from that period to now. What's changed? Is Googlebot getting trapped somewhere? Has crawl traffic dropped?
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Finally, I wouldn't engage in old-fashioned link building. It's a terrible idea for a site with a low DA. If you want to acquire citations, you're going to have to do it by making customers really happy, offering great information and content, and generally offering a great experience.
I hope this helps. There's no easy answer here. You're going to need to take a very strategic approach, rather than focus on a single tactic, if you're going to make this work.
Ian