Here is an explicit way to construct a small modification.

Consider the points $p_1,...,p_{n-3}$. We have $n-3$ codimension two linear subspaces $H_{i_1,...,i_{n-4}}^{n-5} = \left\langle p_{i_1},...,p_{i_{n-4}}\right\rangle$. For any choice of $i_1,...,i_{n-4}$ we define $\{j_1,j_2\} = \{0,...,n-3\}\setminus\{i_1-1,...,i_{n-4}-1\}$. Then, the projection from $H_{i_1,...,i_{n-4}}^{n-5}$ is the rational map
$$
\begin{array}{lccc}
\pi_{i_1,...,i_{n-4}}: & \mathbb{P}^{n-3} & \dashrightarrow & \mathbb{P}^1 \\
& \left[x_0:...:x_{n-3}\right] & \mapsto & [x_{j_1}:x_{j_2}]
\end{array}
$$
We get a rational map
$$
\begin{array}{lccc}
g: & \mathbb{P}^{n-3} & \dashrightarrow & (\mathbb{P}^1)^{n-3} \\
& x=\left[x_0:...:x_{n-3}\right] & \mapsto & (\pi_{1,...,n-4}(x),...,\pi_{2,...,n-3}(x))
\end{array}
$$
Note that the hyperplane $W = \left\langle p_1,...,p_{n-3}\right\rangle = \{x_{n-3}=0\}$ is mapped by $g$ to the point $q_1=([1:0],...,[1:0])\in (\mathbb{P}^1)^{n-3}$. Furthermore, this is the only divisor contracted by $g$. Therefore, blowing-up $q_1\in (\mathbb{P}^1)^{n-3}$ we get a small transformation $g_1:X_{n-3}^{n-3} = Bl_{p_1,...,p_{n-3}}\mathbb{P}^{n-3}\dashrightarrow Y_1^{n-3} = Bl_{q_1}(\mathbb{P}^1)^{n-3}$ mapping $\widetilde{W}$ (the strict transform of $W$) to the exceptional divisor $E_{q_1}$, while the exceptional divisors $E_{p_1},...,E_{p_{n-3}}$ are mapped to the strict transforms of the $n-3$ divisors in $(\mathbb{P}^1)^{n-3}$ obtained by fixing one the factors.

Furthermore, $g([0:...:0:1]) = ([0:1],...,[0:1])$ and $g([1:...:1]) = ([1:1],...,[1:1])$. Let $\mathcal{U}\subset X_{n-3}^{n-3}$ and $\mathcal{V}\subset Y_1^{n-3}$ be the two open subsets on which $g_1$ is an isomorphism. Now, by applying the universal property of the blow-up we get that $g_{1|\mathcal{U}}$ lifts to an isomorphism $f:Bl_{p_{n-2},p_{n-1}}\mathcal{U}\rightarrow Bl_{q_2,q_3}\mathcal{V}$. Since $g_1$ is an isomorphism in codimension one we conclude that $f$ induces a small transformation $f:X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^n\dashrightarrow Y=Bl_{q_1,q_2,q_3}(\mathbb{P}^1)^{n-3}$ mapping $E_{p_{n-2}}$ to $E_{q_2}$, and $E_{p_{n-3}}$ to $E_{q_3}$.