CorrFunction is compiled and thus converts all its inputs into machine precision and also returns numbers in machine precision. All calculations depending on the output of CorrFunction will also be coerced to machine precision.

In order to obtain higher precision results, replace CorrFunction by a conventional function that can take advantage of arbitrary precision numbers. For example with

CorrFunction[θ_] = CorrCurve[θ, ϵ];

I obtain results with Precision close to 95..

CorrFunction is compiled and thus converts all its inputs into machine precision and also returns numbers in machine precision. All calculations depending on the output of CorrFunction will also be coerced to machine precision.

In order to obtain higher precision results, replace CorrFunction by a conventional function that can take advantage of arbitrary precision numbers. For example with

CorrFunction[θ_] = CorrCurve[θ, ϵ];

I obtain results with Precision close to 95.. And the execution of GaussQuadInt[Func, N[1/10, 100], 1, 60, 200] does not take longer than before.