You can try the KCC farmer's market on Saturdays. It's not too far from Waikiki. There is tons of produce there. I saw lychee there in July. I think I the tail end of lychee season is late October, but I'm not an expert.

Anyways, the farmer's market is a definite can't miss place to visit in Oahu there are good eats all over the place. My favorite was the grilled corn and fresh lemonade.

http://www.hfbf.org/farmersMarketKCC.shtml