When I originally filed for my name change, I only ordered one copy of the court order. It turns out that's highly inconvenient when you need to change your name in many different places.
If your court order was processed in Multnomah County, you can request additional copies through the Multnomah County Circuit Court, and there are directions on their "Court Records" page. What it doesn't make clear there, however, is that in addition to requesting records in-person and via mail, you can also request your records via email.
To do so, fill out the Records Request Form and email it to email@example.com which is listed on Multnomah County's official site. Include the mailing address you would like your records sent to, either on the Records Request Form or in your email.
Within a couple days, someone from the records office will respond (via email!) and give you directions for either sending a check or providing your credit card information over the phone during their normal business hours. Once the fee is paid, they will then mail your additional copies via USPS.
In my experience, it took one business day (Thursday) for them to respond and for me to provide my credit card information over the phone. I received my copies in the mail the following Tuesday.