From the west coast, the lowest I've been seeing is around $300 roundtrip, for travel midweek in both directions; weekends will bump that up to the mid-$300s to $400 roundtrip. The "low" season tends to be when most kids are in school, so between about February through the first week of June, and September through early November. From the midwest (like ORD) or east coast (e.g., BOS, JFK, LGA, EWR), I've seen low fares in the low to mid-$500s.
I'm not aware of any flights to Hawaii out of BNA, so you'll need to take a positioning flight, maybe to ORD, LAS, or somewhere in California. Hawaiian flies out of Las Vegas, and SFO, OAK, SJC, LAX, SMF, SAN, and LGB in California. Southwest flies out of SJC and OAK right now. SFO and LAX will have most/all? of the major carriers.
You can set up fare alerts on sites like Google Flights, Skyscanner, or Kayak. You can also check blogs like theflightdeal .com. TPG seems to pull a lot of fare deals from that site, but they have more visibility.
ETA: These are cash fares, which you can obviously redeem for with UR points. I haven't done a comparison between the different points programs in a while so I can't really say, and we tend to stick to Hawaiian or United for our flights since DW has a pile of HawaiianMiles and I have a lot of United miles with SFO being a United hub.
ETA2: Here's a recent article from SFGate's resident travel writer about searching Google Flights and Southwest:
https://www.sfgate.com/travel/article/How-to-find-cheap-fares-to-Hawaii-13705557.php