Thank you for the replies to my questions. I'm just going to start jotting things down here:
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Your client is outside of the Austin city limits. The location is the quite a bit west of the city boundary, as perceived by Google: https://www.google.com/maps/place/Austin,+TX/@30.3172704,-97.91144,11z/data=!4m13!1m7!3m6!1s0x8644b599a0cc032f:0x5d9b464bd469d57a!2sAustin,+TX!3b1!8m2!3d30.267153!4d-97.7430608!3m4!1s0x8644b599a0cc032f:0x5d9b464bd469d57a!8m2!3d30.267153!4d-97.7430608
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I mention this, because Google is fairly biased towards physical location, and a business outside of city borders is not one I'd typically expect to see ranking well locally, unless there was little competition. Organically, I might see them ranking however. So, I'm walking into this not really expecting your client to rank well locally, and I just want to mention that.
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Using BrightLocal's rank checker tool and setting my zip to the zip code at Google's perceived center of Austin (78701) I am unable to find your client in the first 10 pages of the organic results for "austin plumbers". You mentioned you are seeing them on page 9 from another tool. I just can't find them at all.
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Yet, when I search from my own location in California for this same term, your client is #1 in the local pack and #2 organically (just below Yelp). So, for non-Austin-based searchers,your client is doing extremely well. And, this comes as a bit of a surprise to me, given what I said above about them being beyond city limits. To have overcome this, the site must be doing something right to be so dominant!
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So, now, the frustrating part of this. It has long been observed that Google handles proximity of the searcher quite differently when the searcher is near a business vs. in some other city or state. In other words, if, from my location in California, I look for a plumber in Austin, I'm not likely to see the exact same results as someone actually in Austin would see. Rank checking tools aren't great at approximating this. So, I want to ask you to ask your client to do something. Have them search from their office computer on Ranch Road for "austin plumbers" and tell you exactly where they see themselves both locally and organically. Please, if you can do this, come back and let me know what they say.
Clearly, the launch of the new site design is making you question whether something you did caused the ranking drop. But, I want to be sure that we aren't overlooking that Google somehow made a change that is coincidental to your redesign and has nothing to do with it. If your client's site was penalized, I wouldn't expect to see it ranking so excellently for my searches from California. And the GMB listing is not suspended or anything like that. So, let see if you can get your client to tell you what they see searching from their own office. I'll stay tuned.