Thanks very much Zoomkat & PaulS. Both helpful comments.
It turns out that my issue was something much more elementary (and embarrassing). I reviewed SurferTim's code in Reply #17 back in April, and noticed something in his code that mine didn't have:
Serial.println("disconnecting.");
client.stop();
Once I added that to mine, it's worked 100% of the time. 
Priot to adding that, I was creating connections but never closing them, so when subsequent attempts to connect were made, they only worked once in a while (presumably after the original, successful connection had closed because it just timed out!).
Thanks guys!!