A lot might depend upon how the negatives are currently cataloged and stored. Are they in translucent sleeves that allow low-quality scanning without removing them from the sleeves? Or are they in paper sleeves/envelopes? Is there any type of systematic cataloging naming or numbering scheme already in place that would make retrieving negatives selected for later high quality scans easy and efficient?
Since these are 35mm (135 format) negatives, we can probably safely assume that they are cut into strips? My advice would be to use a cheap film scanner, preferably with an automated feed if you can find one, that scans each negative as a separate image. (Plustech also sells extra film holders, so that you could load one while another is being scanned.) If the negatives are not already cataloged you could also place scanned negatives in a cataloging system as you remove them from the scanner and match catalog numbers to image file names/folders used in the scanning process. Even the simplest programs with such scanners will usually allow you to use your own naming/numbering scheme to at least partially match file numbers and folder names to existing catalog numbers.
If you do decide to go the iPhone route, it might be worth it in terms of speed/efficiency to use a device like the Lomography Smartphone Film Scanner. It comes with an app that has both iPhone and Android versions, though most users report that other apps work better for them.
I'd be leery of outsourcing any historically significant material unless a contract is enacted that specifies both liability for any possible physical loss/damage to the negatives as well as liability if confidentiality is breached and the images are made public via unauthorized channels.
The expense of outsourcing 20,000 full resolution scans may be an appreciable percentage of, or even more than, what you are being paid for the job, too. Outsourcing only a limited number of scans needed for eventual publication or exhibition (either digital or print) would probably be more efficient. Even then, you're probably going to need to find some sort of arrangement where you can work with the parties doing the scanning to insure cross-compatibility between your catalog naming/numbering system and the file names of the scans, depending upon how many images are ultimately selected for high quality scanning.