SSN wouldn't help
Due to changes in temperature at different depths, it's unlikely that the sound of a beacon a few thousand meters deep would be detected by a submarine at a couple of hundred meters depth. The sound tends to radiate at about the same depth, which is good in that it doesn't decay as fast horizontally, but in this case would keep the sub from hearing it. Nice idea, but it just wouldn't work.
This is called the thermal layer. All you need to go is beyond the thermal layers that isolate. At some point, the temperature are no longer significant as the temperature will stay the same throughout the year.
e.g. here in Switzerland the lakes tend to be all year the same temperature at a depth of around 20m. The lake stays at 5°C. At the surface diving down it can change
In winter times you have from 3°C at the surface down to 20m and increase in temperature. In summer at the surface to 3m you get around 20°C decreasing as you go down. In summer you have drastic jumps.
The ocean follows the same principle.