We study the generalized covering radii of binary primitive BCH codes, which measure how many parity-check columns suffice to span several prescribed syndromes. For the four-error-correcting family of length $2^m-1$, the second radius is exactly $11$ for $m\geq55$, and it is either $11$ or $12$ for $m\geq16$. For every fixed error parameter at least two, we give an explicit stable bound with two adjacent possible values for the second radius and an arithmetic criterion for exactness. At every higher order, we obtain explicit lower and upper bounds whose additive gap is bounded independently of the order for each fixed error parameter, once explicit field-size conditions hold. The bounds are therefore asymptotically tight as the order grows. The higher-order upper bound retains the common-core support count established by Xiong, Yip, and Zullo; our refinement reduces its sufficient field-size threshold at large order. The refinement combines componentwise mixed degrees with the ordinary degree of a multicone. The upper bounds come from constructing syndrome representations with a common set of parity-check columns.

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