Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense BCH codes $\mathcal{C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^m$ and established that the codewords of the minimum (or the second minimum) weight in these codes support infinite families of 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$-designs with $t=3,4$. In particular, we prove that the codewords of weight $7$ in $\mathcal{C}_{(q,q+1,4,1)}$ support $4$-designs when $m \geqslant 5$ is odd and $3$-designs when $m \geqslant 4$ is even, which provide infinite classes of simple $t$-designs with new parameters. Another significant class of $t$-designs we produce in this paper has supplementary designs with parameters 4-$(2^{2s+1}+ 1,5,5)$; these designs have the smallest index among all the known simple 4-$(q+1,5,λ)$ designs derived from codes for prime powers $q$; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL$(2,2^{2s+1})$ as automorphism group constructed by Alltop in 1969.