Flying might be your best bet, after all...
I'll give you my example- I am a bit short of the 100,000 MQM bonus line. So this is my plan:
Your ingredients: i. Free afternoon, ii. 398 dollars or less, iii. a book or MP3 player (you will spend time on the plane).
(a) Registered for the double MQM from LaGuardia to Boston (BOS) or Washington, DCA (The shuttle promotion)
(b) Ideally, use your Amex (double miles) card.
(c) Buy two RT tickets: LGA-DCA and LGA-BOS. Make sure that the departure for the one you select as the second RT is at least 30 minutes after the first one land (i.e.LGA to DCA 4:00pm-5:10, then DCA-LGA 5:45pm to 7:00 then LGA-BOS 7:10pm-8:20pm then BOS-LGA 9:00-10:15pm). Times vary abit by the day of the week.
(d) Did that? great, go and fly- this is what you'll get, adjust bonus miles by your current status.
each segment: 500 miles and 500 MQM (total 2000 miles and 2000 MQM).
double promotion bonus: 500 MQM per segment (total 2000 MQM).
double miles for status (GM/PM): 2000 miles, if you are silver is 25%, make it 500 miles bonus.
used your amex: 796 miles
All together: 4000 MQM, 4796 miles.
Price per MQM/Mile earned: MQM 0.0995 per MQM, mile 0.08298 per mile.
Yes, other people can show you routes that may even cost less per mile. But, this one can solve your problem in one free morning or afternoon (and if you don't really care about the total number of miles, one RT segment is enough for you to reach your mileage status line)