How do we estimate the dimensions and mass distribution of a galaxy's dark matter halo, beyond the effective limits of its visible matter?

Current research suggests that the Milky Way's dark matter halo may extend up to about 10 - 15 times the size of the visible galaxy, and is believed to be roughly spherically symmetrical. But I don't understand how we deduce that.

In the case of a spherically symmetrical matter distribution, such as the classic "planet with a hole drilled through it", the matter within the radius of the object acts as a single mass at the centre, and the matter outside that radius makes no net contribution to force or motion. So if we somehow stood in such a shaft drilled through the Earth, and had no idea how far above us the shaft extended, we couldn't deduce the mass of any thin shell of matter at distance D from Earth's centre, because the net contribution would cancel out whatever that mass was. So we couldn't conclude much about Earth's higher-up mass distribution, or where this became effectively zero. (We also couldn't deduce anything from depth if we couldn't measure ambient pressure, which might help.)

By analogy, I don't understand how we can draw conclusions about the extent of matter outside a star or region's galactic orbit.

So how can we deduce the mass distribution, or effective extent/size, of a galaxy's dark matter halo, in a "thin shell" region of space far from the galaxy's centre, if there are no visible stars of that galaxy to provide evidence of the local force of gravity at distances beyond the visible galactic extent?

My present understanding, for info:

I understand in overview some of the principles about a galaxy's dark matter halo, including why we believe dark matter is present and such a large proportion of a galaxy's mass, the need for extra mass to explain how a galaxy retains cohesion/rotation, and the fact that unlike visible matter, dark matter is limited in how dense it can get (Virial theorem? of which I think I may understand the conclusion and recognise the name, but nothing more?). I also understand that given a star (or galactic region's average) motion, we can deduce how much mass lies within that orbit, but I would naively expect a symmetrical "thin shell" of matter outside that orbit to have zero net contribution? But while this seems applicable, I'm less sure.