Consider a k-faced null hypothesis, the jth subtest P-value being P_j. Bonferroni corrections (BC), when needed, are wasteful of power. Our present aim is to suggest a method of improving power by exploiting the subject-matter context: we deemphasize those subtests that provide little hope of revealing something interesting because the standard error (SE_j) is large compared with the largest effect that might realistically exist (K_j). The standard BC procedure gives an Overall P-value, or Overall Attained Significance Level, OASL_BC = k*min_j{P_j}. As an extension, let f_j(.) be an increasing, preferably continuous, function through the origin, and g_j(.) its inverse. When applied to summary statistic T = min_j{f_j(P_j)}, the Bonferroni inequality implies P₀{T =< t} =< SIGMA_J g_J(t), so the associated OASL = SIGMA_J g_J(min_j{f_j(P_j)}). This OASL reduces to OASL_BC when the transform functions are all the same (or k = 1). Now, for some index j, the context may indicate that the realistic alternative to H_0j: m_j = 0 is H_(alt)j: 0 < m_j < K_j, K_j being small relative to the associated SE_j. In most models the likelihood ratio, LR_j(data y) = p{y|H_0j} / sup_(alt)j{p{y|m_j}}, is a natural starting point for combined inference and implicitly is a monotone function of (the conventional one-sided) Pj. This suggests defining T = min_j{LR_j} and using the extended Bonferroni rule. The (one-sided) BC procedure is reproduced as long as K_j's are effectively infinite. As K_j/SE_j approaches zero for a certain part of the j's, i.e. as the hope of useful information about these m_j's dwindles, the procedure focusses on the remaining, fewer, subtests, thereby recuperating power. The extended Bonferroni scheme can be built into sequential rejection schemes.