Not easily. Ideally you'd install a smallish amp (2 x 40wrms) to run the new speakers from, but Audi make it particularly difficult to do this.
To get a feed for the amp off the stock HU you'd need to use a wiring adapter as the HU does not use standard RCA's. The adapter then replaces the Audi connector which currently feeds the rear amp/sub, so you then need an additional adapter in order to plug in the rear amp wiring as well as the new one, which is a bit of a pain.
The easiest way to get the stock system to sound better is to relieve the cabin speakers of any Bass, by installing a proper sub (either a stealth sub or conventional box in the boot). You not only then get decent bass from the system, but you can run the HU on minus Bass settings which will provide more clarity and less distortion from the cabin speakers, enabing them to run at higher volumes without distortion.
A stealth sub is particularly good as it's completely hidden, replacing the stock tupperware sub with the amp hidden behind the rear speaker panel, so the system looks completely stock.
I tried everything in the A3 to get a reasonable sounding system, even going aftermarket with the HU, but the 4" front components on their own will never produce quality sound as they are mounted to a flimsy door card loosely clipped to a metal door with absolutely no space. The 6.5" rears have loads of space but again are mounted to a sheet of hardboard which limits the performance of anything you install.
I upgraded the fronts, then upgraded the rears, then changed the HU, then amped the cabin speakers, but all along the problem was the sub. I added a 300w Alpine 10" sub and this sorted everything. Later needed to reclaim the boot space so started trying to upgrade the stock tupperware sub. Installed a Bose sub, which was better but still used the same plastic enclosure so the bass was woolly and flatulant, which eventually led to building a custom MDF enclosure housing a JL Audio sub. In theory this shouldn't work, the 18mm MDF enclosure is rock solid but way too small and the sub is only 6.5", but the transformation is dramatic. Good clean punchy bass with clarity from the front components. You could spend £120 on new front components with little improvement or £300 on a stealth sub and transform the whole system.