So, we got to try again today! I got up at 5:30am and we were out hailing cabs between 6:30 and 6:40. Unfortunately, we were unable to catch one until after 7:00am. The population in Jinan keeps increasing (of course) but the government hasn't approved any new cab licenses for a few years, so it's pretty impossible to grab one during rush hour (which is perpetual).
Long story short, despite traffic, we all made it to the immigration office in time. Mostly we sat around waiting, then we were called in to talk to an officer. Except there was no interview, they just flipped through our paperwork for a few minutes, entered data, took our pictures and dismissed us. Kelly and Ray told us that they used to ask you the usual "Where are you from?" "What's your occupation?" etc. I suppose with so many new foreigners, they're trying to hurry the process along. I'm not complaining, it was all quite painless.
The only bummer is that they'll have our passports for 15 business days (because of the national holiday that means we won't get ours back until October 21st). It's illegal for foreigners to travel within China without their passports (copies won't do), so we'll be staying put until then. Oh well, we'll get to know Jinan pretty well in that time!