I do the same.
Go to a private clinic and opt for government stock subject to availability.
The bigger clinics eg at dischem and clicks seem to get more stock and their consult / injection fees are cheaper, but the personal care you get out of a private clinic for advice on baby, care, sleep, feeding etc could be worth the cost of the consult fee if you are after some support and not just the vaccinations.
First time around I went the private clinic route, it was a tremendous route.
This time, for financial reasons I have gone with a clinic at a pharmacy to save money. I phone before hand (A few days) to check on stock levels, and this clinic does keep aside for a day or two, then I just go private for the MMR and Chicken Pox and 10week Previnar which government doesnt subsidise anymore.