How can I automatically send an SMS to customers when their ordered part arrives?

You would need to utilize job tags and another texting platform. I know with Chiirp, you can do something like that. Pretty sure you can do it with go high level also. Jobber does so much, but those job specific automations need to be done by another program. As far as I know, Jobber's automatic texts are mostly for sales and appointments. You may be able to mess around with a certain type of appointment and a tag, but that's the only thing I can think of. 

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Nice, I think you're heading in the right direction. Once you connect Jobber in Zapier and start building a Zap, it becomes much clearer what triggers are available and what you can build around them. Hopefully the tag or field update works for your setup.

Maybe a work around would be an "after the fact" automation. Once you have changed a status to "part in" then it sends a message to the client? Or maybe add it as a note and when a note is added, then its like a conversation thread with a client that tells them they have a new message?

You might be able to solve this with Zapier by using a tag or custom field in Jobber as the trigger instead of relying on a note.

For example, when the parts arrive you could either add a tag like “Parts Received” or update a custom field from something like “Parts Ordered” to “Parts Received.” Zapier can watch for that change and then automatically send an SMS notification to the client.

In my experience tags and field updates tend to work more reliably as triggers than notes.

If you’re not sure what Jobber events Zapier can trigger from, one trick is to start creating a Zap and connect your Jobber account. Zapier will show you the available triggers, and then you can build your workflow around those.