One thing that very few people seem to know about but is fantastic for visiting Japan is the Seishun 18 ticket (青春18). A lot of people think that it's only for college students, but actually anybody of any age can use them.
For those that don't know it's a ticket that costs about 2400 yen a piece and allows you to travel on ANY JR trains (yes, that means JR West, Jr Tokai, JR East, JR Hokkaido etc...) for a day. You cannot take any trains that requires express tickets so 特急 and 新幹線 are a no go. Anything else is up for grabs. You get the ticket stamped at the station and then you can ride for as long as you want and best of all get off at any station you'd like as many times as you want. It's fantastic for traveling on the cheap and I've used it many times to go as far as from Kyoto to Tokyo (over 550km of rail). The ticket expires after midnight (you can still ride in a train after midnight, you just can't get into another line after midnight).
There are two caveats. First it is still a ticket designed for students so the dates when the tickets can be used are limited to specific periods that coincide with holidays. I posted the dates below (right now is the perfect time to use your Seishun 18 tickets!). Second caveat is that you have to buy them in stacks of 5 which will cost 5x~2400 yen. The tickets are limited to one specific season so you cannot use two in the spring season and then expect to use them later in July. To remedy this you can either buy or sell partly used tickets at local Kinken shops or on Yahoo auctions.
春季
発売期間：2月20日 - 3月31日
利用期間：3月1日 - 4月10日
夏季
発売期間：7月1日 - 8月31日
利用期間：7月20日 - 9月10日
冬季
発売期間：12月1日 - 12月31日
利用期間：12月10日 - 1月10日
The longest distance that's feasible from the first train to the last is usally around 1000km in one day. Obviously if you just want a day trip you won't go that far, but the ticket is worth using for anything over 1200 yen one way. So you could leave at 6AM and get to Nara at 9AM.
発着時間：06:09発 → 09:09着 所要時間：3時間0分 乗車時間：2時間47分 乗換回数：3回 総額：2,590円 距離：159.0km
■岐阜
｜ 東海道本線（東海）(大垣行) 49.6km
｜ 06:09-06:20［11分］
｜ 2,590円
◇大垣 ［4分待ち］
｜ 東海道本線（東海）(網干行) ↓ 前～後
｜ 06:24-06:59［35分］
｜ ↓
◇米原 ［5分待ち］
｜ 琵琶湖線新快速(姫路行) 67.7km 前・後
｜ 07:04-08:05［61分］
｜ ↓
◇京都 6番線着・10番線発 ［4分待ち］
｜ 奈良線区間快速(奈良行) 41.7km
｜ 08:09-09:09［60分］
｜ ↓
■奈良
Navitime tells me even with ETC that a trip to Nara would cost 4260yen each way and take about 30 minutes less. So it's about 4 times as expensive as using the Seishun 18 ticket!
One important thing is to always check the train schedules on Jorudan or Yahoo rosen. It's crucial to have good connections if you don't want to waste too much time.
Maybe I'm biased but I recommend you go to Kyoto again because it's only 2 hours by JR and there are tons of things to do outside just the main temple attractions.