I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have been so far, only searching for quadratics, searching up to a depth of only $10$ primes initially with $$\frac1d \prod_p\frac{1-\frac{n_p}{p}}{1-\frac1p},$$ where $n_p$ is the number of roots of $f(x)$ modulo $p,$ and $d$ is the degree of $f(x)$, with values over $3$, and then refining. I have then been using the quadratic patterns from here with coefficient patterns $+,-,+$ as per the recodrs published therein.
However, I have only been getting max $28$ prime-runs (aside from the expected longer runs associated with the Heegner numbers), searching $10^5$ interations, and then up to $10^6$ $n$ for each quadratic (which is no doubt the downfall, since the resultant runs appear to increase (as expected) the higher the search limit.
Is a more systematic approach more likely to reap better results, or is there a far more efficient method for searching?
NB Coding examples can be provided if desired, but are surely not optimal.