I'm able sometimes to fruit the same Star Brite plants from late June until October. But it takes constant care. The right amount of fertilizer, water, and usually foliar disease control.

You are right in observing the fact that they have to keep growing in order to flower and fruit. In addition the leaves need to stay healthy and/or grow enough new to make a sweet fruit.

It's easier here to have at least two plantings. One for early fruits and one for the late summer/fall.

Star Brite is my favorite. It averages more than 25 lbs on vigorous vines and runs 11-14 brix, 14 is almost too sweet. Seedless are too hard to grow, have smaller fruit, and don't produce as much. I've tried many others and Star Brite wins every year. But melons are very regionally adapted. What works here won't look as good elsewhere.

I am trying lots of Heirloom varieties this year. They actually are growing better than the storebought common varieties...
Here is what im growing:
http://forums2.gardenweb.com/forums/load/cornucop/msg051103452232.html

From alot of reviews I'd say, next year try these:
Ali Baba (its light and doesn't sunburn... also it grow fastest for me even in this humid/wet summer)
Moon and Stars (pretty and dependable for me in this wet summer)
OrangeGlo (supposedly one of best flavored, moving them in a truck supposedly they break/crack... so only for small scale home gardener... but i really want to try one)
BlackTail Mountain (just overall very good reviews)

The worst ones growing for me were the common store ones: Sugar Baby, Crimson Sweet, Fairie. Very slow growing for some reason ... Heard Sugar Baby is usually good one though.

I think I only watered lightly a couple times whole summer to get nutrients into soil and foliar spray. just been getting so much rain, i dont want them to turn less sweet by watering too much.