Multiple stone throws

A country mile means a fair distance; for some reason, a country mile is thought larger than other kinds of miles. This source here has it as a "great distance", but I don't think that's quite right. A country mile is a fair step, not a long way.

This doesn't follow your "widely recognised" criterion, but I thought I would mention in Australia the phrase "within cooee" is similar to a "stone's throw", but the distance would be much greater. It is supposed to be the distance at which you can hear another person bellowing the long oo and ee vowels.

From Wikipedia:

Cooee! (IPA /ku:'i:/) is a shout used in Australia, usually in the Bush, to attract attention, find missing people, or indicate one's own location. When done correctly - loudly and shrilly - a call of "cooee" can carry over a considerable distance.

...

An expression "within cooee of" has developed. It means "not far from", and seems to be confined to New Zealand and Australian English, and is often used in the negative sense (ie "not within cooee of" meaning not close to).

What about an obvious: a few stone throws away?

There is some complexity in imagining, and a lot of uncertainty, of how far that actually would be; do note that even the term "stone throw away" is very often used metaphorically - for distances that are obviously further away than stone throwing range, but are comparatively short in the given context.

Such use suggest that there is no common term that would be useful, however you always have other options, for example:

• 20 minutes away, by { insert mode of transport: foot, car, bicycle, etc }

which can be considered intermediate distance expressed in commonly understandable way.