The textual layer has Rama, before building his bridge to Lanka, tell Sita how Shiva appeared and was worshipped as a linga on the island; the Skanda Purana's Setu section and the Shiva Purana have him propitiate Shiva with mantras, meditation and dance, receive the promise of victory over Ravana, and ask Shiva to stay on the island to sanctify the world.
Base telling: worship precedes the war; no sand linga, no Hanuman errand.
The temple's own story places the worship after the war. Having killed Ravana, a brahmin and a Shiva devotee, Rama is told to expiate the killing by worshipping Shiva, and sends Hanuman to Kailasa for a linga. When the auspicious hour draws near and Hanuman has not returned, Sita shapes a linga of sand, the Ramalinga, and it is consecrated. Hanuman's linga, the Vishwalinga, is installed beside it and, to console him, is worshipped first to this day. The shrine counts among the twelve Jyotirlingas and the Tamil Paadal Petra Sthalams.
Worship follows Ravana's death as expiation; adds Sita's sand linga, Hanuman's Kailasa linga and the order of worship.
Facts about Ramanathaswamy Temple — Vedicfeed; Ramanathaswamy Temple — Wikipedia
On the banks of the Krishna the same story is told of two shrines facing each other. Rama, returning north, installs lingas as he goes; Sita's linga of sand keeps growing until she caps it with a knot of sand, and Hanuman's linga from Kailasa is set on the opposite bank. The Skanda Purana is cited as the authority.
Relocates the episode to the Krishna river on the return journey; the sand linga grows and must be capped.
Sources: Ramanathaswamy Temple — Wikipedia; Facts about Ramanathaswamy Temple — Vedicfeed; Ubaya Rameswara Kshetram — Wikipedia
Ravana and his brothers Kumbhakarna and Vibhishana perform austerities at Gokarna for ten thousand years.
In the Tamil Ramavataram Rama is divine from the start, yet the poem sets him above Brahma, Vishnu and Shiva rather than inside the triad as an avatar.
A tribal woman finds Rama's images in an anthill on the Godavari; a tax official builds the temple with state money and is jailed for it.
A fish warns a brother and sister of a flood; afterwards Bhagwan, or in one telling Rama, sends them to repeople the world.
Thailand's Ramakien makes Phra Ram an incarnation of Phra Narai and a Bodhisatta; Cambodia's Reamker reads Preah Ream almost as a mortal through a Theravada lens.
Jain tellings keep Rama and Krishna as great men in a fixed roster of sixty-three: Lakshmana kills Ravana, Rama becomes a monk, and Krishna is Neminatha's cousin.
Shaiva texts and popular tellings say that when Vishnu chose to be born as Rama, Shiva chose to be born as Hanuman — his eleventh Rudra — to serve him; Vaishnava schools call Hanuman the son and incarnation of Vayu.
Rama wishes to worship Shiva on the island before crossing to Lanka, or after killing Ravana, to be cleansed of killing a brahmin.