Ok, from some Qantas brochures and maps I sent away for in about 1966/67 when I was just an airport kid here's some info that may help.
Ordered in 1956 the first B707-138 was delivered to Qantas in 1959.
"Jet services to the U.S.A. commenced on July 29th, 1959 and continued to Lodon via new York after September."
27 October that year saw the 707 put on the Sydney/London (via India) Kangaroo Route.
The first turbofan B model B707-138 ferried into Sydney non-stop from Honolulu (5,185 miles in 9 hours 46 min) on 6 August 1961.
"The first regular V-jet [from Latin vannus = fan, quite cultured was our QF back then] to San Francisco commenced on September 24 , extending next to London and finally right around the world on October 29th."
The routing was Sydney-Nadi-Honolulu-San Francisco (with a once a fortnightly return flight SFO to vancouver, a leftover from the old British Commonwealth Pacfic Airlines route) - New York - London.
From November 1961 the -138Bs began operating to Hong Kong and Tokyo; and by January 1962 had completely replaced the Lockheed Electras on these routes.
The routing appears to be Sydney - Darwin - Manila - Hong Kong - Tokyo or Sydney - Port Moresby - Manila - Hong Kong - Tokyo; 3 times a week.
Perth was upgraded to international jet standards by February 1962 and "then became a regular stop on two of the four weekly Kangaroo Route Services through Singapore to London."
These routes were Sydney-Darwin-Singapore; Sydney-Brisbane-Darwin-Singapore; and Perth-Djakarta[sic]-Singapore.
Then it was Singapore-Kuala Lumpur-Colombo-Bombay-Teheran; then Teheran - Athens, or Teheran - Istanbul - Athens; Athens - London.
[hmm, seems to be a slight inconsistency with one of the brochures stating that Kuala Lumpur and Colombo were not up to jet standard until September 1965, by which time the first of the -338Cs had arrived.)
Or Singapore-Bangkok, then Bangkok-Calcutta-Karachi-Cairo, or Bangkok-New Delhi-Karachi-Cairo. Then Cairo - Rome - London; or Cairo - Athens - Frankfurt - London.
At that time Electras were used on flights to New Zealand, and had replaced the last Constellation route; Perth - Cocos Island - Mauritius - Johanesburg.
Unfortunately I did have a couple of timetables from that era but I can't find them (not that they would specify -138B or -338C anyway I guess).
On 28 November 1964 the "Fiesta Route", Sydney - Nadi - Papeete - Acapulco - Mexico City - Nassua - Bermuda - London, was inaugurated.
This must have been with -138Bs because the first -338C wasn't delivered until 8 March 1965 (departing SFO the previous day and flying non-stop to SYD in 14 hours 32 min, the first non-stop trans-Pacific flight).
Interesting routes when you look back on them.