Let \(p:{\mathbb N}\to{\mathbb N}\) be the partition function, that is, \(p(n)\) denotes the number of integer partitions of \(n\). The associated Jensen polynomial of degree \(d\) and shift \(n\) is then defined by \[ J_p^{d,n}(x):=\sum_{j=0}^{d}\binom{d}{j}p(n+j)\,x^j. \] A Jensen polynomial is said to be hyperbolic if all of its zeros are real. Recent results show that for each \(d\) there exists some \(N\) such that for all \(n\geq{N}\) the polynomial \(J_p^{d,n}\) is hyperbolic. Define \(N(d)\) to be the minimal such \(N\). In this paper, the authors prove that \(N(3)=94\), \(N(4)=206\), and \(N(5)=381\). Moreover, the upper bound \(N(d)\leq(3d)^{24d}(50d)^{3d^2}\) is proved for all \(d\in{\mathbb N}\).