Only if you don't intend to leave the country at all. Once you cross the border, even if it's a short weekend trip, you would have to apply for a new E-3 visa again.
This can be a big deal. I changed from L1B to L1A to H1B (without leaving the country) then when I wanted to leave the country the legal advice that I received (IANAL - TIJAA[1]) was that to be safe I should return to my home country to get the new visa stamp, the Vancouver embassy might legitimately decide not to apply it - at which point I'd be stuck in Canada needing a plane ticket to the UK in a hurry and have the possibility of a multi-week wait for an appointment at the London embassy...
I waited on my green-card application to get to the point at which I could get an Advanced Parole[2] document and traveled on that instead, but this did mean: a) several years in which I couldn't travel outside the US; b) repeated lectures from border officials that the AP document was for "serious travel" only and that snowboarding trips to Whistler didn't count.
For short trips of 30 days or less, to Canada, Mexico or adjacent islands excluding Cuba, it seems you can re-enter on your expired visa and your new I-94, once your extension of stay has been approved. 22CFR41.112(d)[1]. If it were me I would get a new visa though.
I changed jobs this way recently and it took ~6 months to receive a decision from USCIS.
Others have been done much quicker but I would be prepared to wait going this route, and also take USCIS's processing times with a grain of salt.
If the E-3 allowed for premium processing[1], portability[2] or the 240-day rule[3], I would try I-129 again otherwise I think I'll just apply for a new visa outside the US.
1: Pay an extra ~$1200 to hear back from USCIS in ~2 weeks.
2: Permission to start work for a new employer upon filing the I-129.
3: A 240 day grace period during which you may continue to work for your existing employer after your status expires.
That's actually not true[1]. You can change status to an E-3 from within the US by filing Form I-129.
[1] http://1.usa.gov/boasGi