The following is a consequence of Theorem 1 in "A. Bertram, Moduli of Rank-$2$ Vector Bundles, Theta divisors, and the geometry of curves in projective space, J. Differential Geom. 35, 1992, 429-469."
Let $C\subset\mathbb{P}^{2h}$ be a degree $2h$ rational normal curve. Consider the following sequence of blow-ups:
$\pi_1:X_1\rightarrow\mathbb{P}^{2h}$ the blow-up of $C$,
$\pi_2:X_2\rightarrow X_{1}$ the blow-up of the strict transform of $Sec_2(C)$,
$\vdots$
$\pi_{h-1}:X_{h-1}\rightarrow X_{h-2}$ the blow-up of the strict transform of $Sec_{h-1}(C)$.
Let $\pi:X\rightarrow\mathbb{P}^{2h}$ be the composition of these blow-ups. Then, for any $k\leq h$ the strict transform of $Sec_{k-1}(C)$ is smooth, irreducible and transverse to all exceptional divisors. In particular $Y$ is smooth and the divisor in $Y$ given by the union of the exceptional divisors and the strict transform of $Sec_{h}(C)$ is simple normal crossing.
It is enough to apply Theorem 1 of the above cited paper and observe that The rational normal curve is given by the Veronese embedding induced by the line bundle $L = \mathcal{O}_{\mathbb{P}^1}(2h)$ on $\mathbb{P}^1$. Now, $$h^{0}(\mathbb{P}^1,L(-2h)) = 1 = 2h+1-2h= h^{0}(\mathbb{P}^1,L)-2h.$$ This means that $C\subset\mathbb{P}^{2h}$ is embedded by a $2h$-very ample line bundle.
