How to remove tap water hardness?
Without knowing what dissolved species are causing the hardness, it's a little difficult to know what to advise. Performing qualitative analysis would likely help you a lot.

You describe using lime softening (i.e., using hydroxides) which can help in some cases. Usually the resulting hydroxide precipitates are pretty hard to filter, requiring a membrane or a very good filtration system.

As Wikipedia notes, there are a few other reliable techniques:


reverse osmosis
distillation
ion exchange resins
chelating ligands


While the first three are definitely the best in terms of ultimate water quality, an easy thing to supplement lime softening would be to add various chelating ligands. For example, after you carefully filter the water from lime softening, maybe add citric acid?

In any case, I'd suggest you do some sort of qualitative analysis (and pH) testing as you go. This is the best way to know what steps you'll still need.