Changing the order of fields in an existing geodatabase feature class or table has been challenging since the early days of ArcGIS - ArcMap and ArcCatalog. Without bothering to dig up all of the historical references, blog posts, issue reports, bug reports, great idea postings, workarounds, previous methods and tools, etc, I point at one simple solution to this 16 year old problem (it's old enough to drive in most US states).
- Purchase and install Sparx Enterprise Architect (EA) version 12.x or higher, Professional edition or higher, floating or single use license.
- Esri recommends it: A note about the use of UML for geodatabase design—ArcGIS Help | ArcGIS for Desktop
- It's affordable: Enterprise Architect - Pricing and Purchasing
- if you spend a few hours that your employer bought from you trying and failing to re-order geodatabase fields, your employer has very well paid for the EA software and has nothing to show for it. << justification for your request.
- There are some kickstarter vids: ArcGIS Geodatabase Design | UML | Enterprise Architect
- ArcMap: export your disorderly geodatabase as a Workspace XML
- Edit: Schema only, not the data (thanks Joshua Bixby)
- Tools > Options > Objects > uncheck "Sort Features Alphabetically" > Close
- File > New Project > name it > Model Wizard > click "Geospatial" > check the "ArcGIS Workspace" box > OK
- Extensions > ArcGIS > Import ArcGIS Workspace XML
- double-click the "Workspace" object in the project browser to open a graphic view of your geodatabase schema
- find the disorderly table or feature class graphic
- left-click any of the fields in the table or feature class graphic to select it
- right-click the selected field > View Properties...
- right-click the field that you want to be in a different order and choose "Move up" or "Move down" as pleases you
- Close the field properties viewer
- Extensions > ArcGIS > Export ArcGIS Workspace XML
- import your orderly schema from your Workspace XML into your geodatabase
- load your data into your new schema
- breathe, sigh, rejoice... move along
That's it. That's all I got.