Is there any difference between the two ideas:
"She came from Spencer, across the hill"
"She came from Spencer, beyond the hill"
It sounds to me that across is more near of the hill than beyond, but I'm not sure.
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I'll address on this context and not other uses of 'beyond/across'.
In laypersons' language, 'beyond' has a connection with 'far'. So, if you are telling that someone is going 'beyond' something, you mark the hill as a point to describe what went farther.
OALD example says:
The road continues beyond the village up into the hills. -the village is the mark and the road goes farther into the hills.
On the other hand, 'across' talks about two points and in between is your object. So, if something is coming 'across' the hill, it's passing by the hill or getting 'through' it.
To get across the hill, you need great stamina. It's almost 45 km, you know!
Here, I'm taking two points -from where we are standing and I'm speaking and to cross the hill and go beyond it!
Somehow, I feel that 'came' fits better with 'across'; 'beyond' is generally observed with 'go'. However, there could be exceptions but what I said is 'general'.