On those occasions where we have had to make use of an external contractor 5 weeks+ seems to be pretty much the norm with quite a few being longer than that. It probably depends a bit on where you are, it would seem in the UK for example that a lot of places are currently very busy, that pushes lead times out, especially for smaller batches. To be able to truly turn things around very quickly really requires the contractor to have a line that is not fully utilised, which isn't often a desirable situation.
Kitting the parts yourself could help, provided you make sure you understand how they want their parts kitted, however even then we have seen lead times where the time doesn't even start to count down until that kit enters the building (presumably to avoid the "it's on the way honest guv" lies people might tell to get themselves higher up the queue). Bigger batches can help, or if the contractor offers it - call offs that let them plan well advance for your years/period of choice requirement. What can also help tho' is understanding why 6 weeks isn't good enough, can you adjust your processes to make that work?
Equally it could just be a case of finding a contractor that's a good fit for you who can work to those timeframes. Here we would normally expect to be able to do 3 weeks without too much difficulty where we source everything. While a week or less is technically possible in the current climate its quite hard to do, the timeslot has to be there, the parts have to be in stock on the right continent in the right packaging and the distributor has to ship the part you ordered, something we are currently finding happens less reliably than it should in the case of RS and Farnell, particularly when you are buying quantities that mean cut tape or re-reels, if sods law can strike, it will. I don't think our ability to deliver in those lead times is all that unique, I would expect something similar to be possible in Aus, I would look local and look small, what you are looking for is probably a specialist or someone like us that mixes OEM and contract manufacture.
As an example last month I build 10 each of several fairly massive boards for a company around the corner, they kitted everything because they kit for their volume contractor anyway, I was building those boards 2 days after they send the files over for a quote. When I found kitting errors they were there 20 mins later with the correct part, if that had been a Mouser only that could easily have been a 2/3 day spanner in the works (something very odd happens between the USA and the UK with DHL, parcels frequently seem to miss the plane in France on their tracking) and the jobs we had queuing up behind would have had to take priority, in that instance that would have added 2 weeks (because it would have been their fault).