Plain old distillation.
Now, distillation is expensive. It requires huge amounts of energy. So how much they can produce will depend on just how rich they are, and specifically how much energy they can command. If they have oodles and oodles of easily accessible fossil fuels, they might very well be able to produce just as much as they got from an aquifer.
The simplest set up would be a solar still. A well-designed industrial-sized solar still, with concentrator mirrors, will produce much more drinkable water than passive evaporation, but you'd need a lot of them, over a very wide area to capture enough solar power to make enough water for your whole civilization. The mirrors don't need to be particularly high-tech; in particular, they don't need to be image-forming. Polished brass would be fine. You'd just have to make sure it's well-maintained. Alternatively, you could ditch the mirrors and just spread out the water over a much larger surface area--but that requires building a much larger still, with a much larger transparent roof (basically a greenhouse for water) for letting light in, holding in the heat, and preventing the escape of the vapor. Which may very well end up being a more difficult proposition, not less, especially considering the expense of glass in the ancient world and the technical difficulty of producing large sheets of it. It doesn't need to be optical quality, but still, that's asking a lot.
The other option is fire-powered stills. If they have access to fossil fuels, or a large supply of rapidly-regrowing vegetative fuel sources (like, say, a really big bamboo forest), solar concentrators can be replaced with furnaces to heat water for distillation. In theory, such a still could be built entirely out of ceramic pottery, but any amount of metal or glassworking technology they have would be helpful.
In any case, note that one of the waste products of the distillation process will be highly concentrated brine--and that's not necessarily a bad thing! Ideally, you'd want to keep up a good flow through your distillery so that you don't build up salt depositions inside (or so that you minimize build up--cleaning the works will be a necessary regular maintenance task regardless), and you can pipe the concentrated brine elsewhere. Don't throw it away, though! Put it into passive evaporation pools, and start manufacturing salt! Salt is an incredibly valuable trade good for ancient civilizations, and they could then start using it to buy supplies fro other civilizations--including fuel, and possibly even aqueduct access to other people's water, if necessary.
On top of all that, there are things they can do to reduce their dependence on fresh water. You can, for example, use salt water just fine for evaporative cooling. Additionally, they could try developing saltwater agriculture, as mentioned in Ash's answer, and aquaculture. Sea plants obviously already grow in saltwater just fine, so farming and using seaweeds as a large part of their own diet, as animal feed, or even as fuel for the distillery furnaces, would cut down on how much water they need to distill, as well as eating fish and other seafood to avoid having to spend water on as many domestic land animals.