http://oeis.org/A025018 gives "Numbers n such that least prime in Goldbach partition of n increases," and begins, 4, 6, 12, 30, 98, 220, 308, 556, 992, 2642, 5372, 7426, 43532, 54244, 63274, 113672, 128168, 194428, 194470, 413572, 503222, 1077422, 3526958, 3807404, 10759922, 24106882, 27789878, 37998938, 60119912, 113632822, 187852862, 335070838.
http://oeis.org/A025019 gives "Smallest prime in Goldbach partition of A025018," and begins, 2, 3, 5, 7, 19, 23, 31, 47, 73, 103, 139, 173, 211, 233, 293, 313, 331, 359, 383, 389, 523, 601, 727, 751, 829, 929, 997, 1039, 1093, 1163, 1321, 1427, 1583, 1789, 1861, 1877, 1879, 2029, 2089, 2803, 3061, 3163, 3457, 3463, 3529, 3613, 3769, 3917, 4003, 4027, 4057.
So, for example, looking at the 5th entry in each list, the smallest prime $p$ such that $98=p+q$ and $q$ is prime is $p=19$, and a prime smaller than 19 will work for $2n\lt98$.
So you can use these lists and the references given at the site to work out how big the smallest prime might have to be for $2n$ up to a given value. If that's what you are asking about.