For days to count under a bilateral agreement but not the 90/180 rule, you must have exhausted your 90/180 rule days but not your bilateral agreement days. So you would have to finish your stay in Spain or Austria and leave the Schengen area directly from one of those countries.
Spending your time in Estonia at a later time, therefore, will not be possible.
Note that the Australian embassy in Spain makes no mention of a bilateral agreement. The Austrian government, however, has a fact sheet that strongly implies that any stay under the bilateral waiver must occur after a previous stay in the Schengen area. It also requires leaving and reentering Austria directly (i.e., by air). I could not find analogous information from Spain.
The Wikipedia article links to the 1961 agreement with Spain, which is not explicit about how it interacts with Schengen rules because the Schengen area did not exist in 1961.
The relevant part of Schengen legislation is Article 20(2) of the Convention Implementing the Schengen Agreement. The first two paragraphs of Article 20:
Aliens not subject to a visa requirement may move freely within the territories of the Contracting Parties for a maximum period of three months during the six months following the date of first entry, provided that they fulfil the entry conditions referred to in Article 5(1)(a), (c), (d) and (e).
Paragraph 1 shall not affect each Contracting Party's right to extend beyond three months an alien's stay in its territory in exceptional circumstances or in accordance with a bilateral agreement concluded before the entry into force of this Convention.
Once more, the provision concerns "extending" the stay beyond three months based on the bilateral agreement. It does not seem possible to invoke the agreement earlier so as to be able to stay for an additional period in a country with which Australia does not have a bilateral agreement.
The only way to do this, I fear, is to rearrange your travel to comply with the 90/180 rule, or to get a national visa for Estonia. That would remove the days spent in the Estonia from the 90/180 calculation. The problem is that there may not be a national visa category that is appropriate for your visit, and Estonia is unlikely to entertain the application because you are not planning to be there for more than 90 days. But you can always try. Be certain to mention in your application that you are applying because you need to go to Estonia for a period of stay that would exceed the 90/180 rule. The website of the Estonian embassy in Canberra is at http://canberra.vm.ee/consular_information.
You may also be able to extend your stay in Estonia by applying for a residence permit after you arrive, but I would be extremely cautious about this as there does not appear to be a category of residence permit that is likely to apply to you.