The main contribution of this paper resides in providing novel algorithmic advances and analytical insights for the sequential hiring problem, a recently introduced dynamic optimization model where a firm adaptively fills a limited number of positions from a pool of applicants with known values and acceptance probabilities. While earlier research established a strong foundation -- notably an LP-based $(1 - \frac{e^{-k}k^k}{k!})$-approximation by Epstein and Ma (Operations Research, 2024) -- the attainability of superior approximation guarantees has remained a central open question. Our work addresses this challenge by establishing the first polynomial-time approximation scheme for sequential hiring, proposing an $O(n^{O(1)} \cdot T^{2^{\tilde{O}(1/ε^{2})}})$-time construction of semi-adaptive policies whose expected reward is within factor $1 - ε$ of optimal. To overcome the constant-factor optimality loss inherent to earlier literature, and to circumvent intrinsic representational barriers of adaptive policies, our approach is driven by the following innovations: -- The block-responsive paradigm: We introduce block-responsive policies, a new class of decision-making strategies, selecting ordered sets (blocks) of applicants rather than single individuals, while still allowing for internal reactivity. -- Adaptivity and efficiency: We prove that these policies can nearly match the performance of general adaptive policies while utilizing polynomially-sized decision trees. -- Efficient construction: By developing a recursive enumeration-based framework, we resolve the problematic ``few-positions'' regime, bypassing a fundamental hurdle that hindered previous approaches.