Can we visit Miyama and then continue to Amanohashidate without going back to Kyoto
by trains/buses? Yes, as long as here "Miyama" means an area in Nantan City, Kyoto Prefecture.
(If your news source says like gMiyama Town, Kyoto Prefecture" that may be too old.)
How is your itinerary in Miyama area?
Amanohashidate is not served by JR but served by Kitakinki Tango Railway [KTR].
If you start from Kyoto City early in the morning, I guess you can visit Miyama (in Nantan City) then from Sonobe Station (of JR-West) take a direct Ltd. Express train to Miyazu / Amanohashidate Station (of KTR) in the evening.
Traveling with a nationwide Japan Rail Pass or a JR-West Kansai WIDE Area Pass, you will have to pay separately to KTR.
JR-West will revise its diagram on 15 March 2014. Timetables of buses for traveling Miyama area may be revised in accordance with new JR timetables.
From Kyoto Station (of JR-West) you can take a Local / Rapid train on Sagano Line to Sonobe Station, which may require about 40 minutes.
Nantan City seems to be operating a bus line connecting West Entrance of Sonobe Station and some tourist spots.
(Nantan City buses are also available from Hiyoshi and Wachi Stations, but direct Ltd. Express trains for Amanohashidate skip these stations.)
Please note that only a small number of trains are available between Sonobe Station and Fukuchiyama Station.
If the timing of a direct Ltd. Express Hashidate is too early / late for you, you may well have to take a JR Local train from Sonobe to Fukuchiyama then make a transfer to KTR.
In Kyoto Station building, trains running for Sonobe (including some Ltd. Express trains) use Tracks 31 - 33. The platform for Track 0 [zero] and the platform for Tracks 30 - 34 and joined, and they are near Central Gate and around the north end [Karasuma end]. (Shinkansen tracks of JR-Central are around the south end [Hachijo End].)
In Fukuchiyama Station building, platforms of JR-West (Tracks 1 - 5) and KTR (Tracks 1-2) are on the 3rd floor, and on the 2nd floor there are concourses and a transfer gate.
http://ktr-tetsudo.jp/station/fukuchiyama/img/fukuchiyama_yard.gif
