Workaround:
At first I convert Hypergeometric2F1 to simpler function like LerchPhi with Maple and using AsymptoticSum to expand series s parameter at infinity.
$Version
(*"12.1.0 for Microsoft Windows (64-bit) (March 14, 2020)"*)
M = 100 (*More terms greater precision *)
Block[{$MaxExtraPrecision = 302}, N[(-1 - s)/(E^10 (1 + E^10)) + (s (1 + s) AsymptoticSum[z^k/(
k + s), {k, 0, Infinity}, {s, Infinity, M}])/E^10 /. z -> -Exp[10] /. s -> 10^50, 300]]
(*2.06096648272347233418959215235994959191908111757491936251422339815271\
8558212616552192199567106427341226560922751917789191834346105243809432\
5329127379111913999269570785982677759971629508381569251261807659636488\
5344917825480306093143427886092553493591682250667234723102601737564361\
910230136964528670935*10^-9*)
Plot[{Hypergeometric2F1[2, 1 + s, 2 + s, -Exp[10]], (-1 - s)/(
E^10 (1 + E^10)) + (s (1 + s) LerchPhi[-E^10, 1, s])/E^10}, {s, 0,
10}, PlotStyle -> {Red, {Dashed, Black}}](*Looks the same*)
I found a Issue in AsymptoticSum:
AsymptoticSum[ z^k/(k + s), {k, 0, Infinity}, {s, Infinity, 1}](*Works fine*)
AsymptoticSum[z^k/(k + s) /. z -> -Exp[10], {k, 0, Infinity}, {s, Infinity, 1}](* Gives input !!! *)
Regards M.I.