Next steps:
(0) (Relative Morse Theory) Geoffrey Mess' paper "Torelli groups of genus two and three surfaces" studies some relative Morse theory of the Abel-Jacobi period locus in the Siegel upper half spaces to deduce that the Torelli group (in genus two) is a free group on countably many generators. I thought his proof was very interesting, and tried to learn more, but hardly made progres...
(1) (Almost Complex Structures) if you're interested in symplectic topology, then Eliashberg-Cielebak's textbook "From Stein to Weinstein and back: Symplectic Geometry of Affine Complex Manifolds" has very interesting treatment of Morse theory, especially as related to almost-complex structures $J$ on symplectic manifolds $(M, \omega)$. I think this textbook eclipses Milnor's texts. Contains very elementary proof that "any $2n$-dimensional complex manifold has the homotopy type of an $n$-dimensional CW-complex". (Indeed the CW complex is totally lagrangian with respect to suitable symplectic structure, therefore at most $n$-dimensional).
(2) Gradient flows to poles (where a potential function $f$ and its gradient $\nabla f$ diverges to $\pm \infty$) appears to have much more applications to topology than the conventional gradient flow to zeros. The biggest problem with "gradient flow to zeros" is that the gradient flow slows down as it approaches its target, and oftentimes an appeal to Lowasiejiwcz inequality is necessary to establish the Lipschitz continuity of the reparameterized gradient flow. In my applications of optimal transport to topology, I found gradient flow to poles much more convenient, since the gradient enjoys a finite time blow up, and continuity of the reparameterized flow is immediate without any appeal to Lowasiejiwcz. Basically "gradient flow to zeros" is a soft landing, while "gradient flow to poles" accelerates into the target.
(3) (Optimal Transportation) Morse theory takes on new form in optimal transportation, where Morse theory plays a role in establishing the regularity (continuity) and uniqueness of $c$-optimal transports. Consider a source probability space $(X, \sigma)$, target $(Y, \tau)$, and cost $c: X\times Y \to \mathbb{R}$. Kantorovich duality characterizes the $c$-optimal transport from $\sigma$ to $\tau$ via $c$-convex potential $\phi=\phi^{cc}$ on $X$ with $c$-transform $\psi=\phi^c$ on $Y$. Kantorovich says the $c$-optimal transport plan $\pi$ is supported on the graph of the $c$-subdifferential $\partial^c \phi$. The subdifferential is characterized by the case of equality in $$-\phi(x)+\psi(y)\leq c(x,y).$$ Differentiating the case of equality with respect to $x$ and $y$ yields the equalities $$-\nabla_x \phi(x)=\nabla_x c(x,y)$$ and $$\nabla_y \psi(y)=\nabla_y c(x,y).$$
If $Y\to T_x X$ defined by $y\mapsto \nabla_x c(x,y)$ is injective for every $x\in X$, then $$y=T(x):=\nabla_x c(x, \cdot)^{-1}(\nabla_x \phi(x))$$ defines an optimal map. Moreover the fibre $T^{-1}(y)$ can be characterized as the set of $x$ satisfying $\nabla_y\psi(y)=\nabla_y c(x,y)$.
Using the usual Implicit Function theorem, the fibre $T^{-1}(y)$ is a smooth submanifold of $X$ if $D_x(\nabla_y c(x,y))$ is nondegenerate for every $x\in T^{-1}(y)$. If the target $(Y, \tau)$ is one-dimensional, this requires the function $x\mapsto \nabla_y c(x,y)$ to be critical point free for every $y\in Y$, and $x\in T^{-1}(y)$. On most source manifolds $(X, \sigma)$ it is difficult to verify the nonexistence of critical points. If $X$ is compact, then Morse theory (elementary calculus) forbids it.
Furthermore, Morse functions are known to be generic (in sense of Sard, Thom, etc.). But we want canonical Morse functions. From mass transport perspective, we seek canonical cost functions $c$ whose derivatives $\nabla c$ are suitable Morse-type functions. For example, if you want to study optimal transportation from a closed surface $\Sigma$ to the real line $\mathbb{R}$, then one seeks an appropriate cost $c: \Sigma \times \mathbb{R} \to \mathbb{R}$ satisfying the above conditions, e.g. that $\frac{\partial c}{ \partial y}(x ,y)$ be critical point free in $x\in \Sigma$ for every $y\in \mathbb{R}$. This is forbidden by Morse theory if $\Sigma$ is compact and $c$ is everywhere finite. (In applications, we allow $c$ to have $+\infty$ poles. Then $\partial c/\partial y$ is possibly critical point free). But the very important question remains: What is a canonical cost $c: \Sigma \times \mathbb{R} \to \mathbb{R}$ which represents an interesting geometric transport from $\Sigma$ to $\mathbb{R}$? Since the source and target spaces $\Sigma$, $\mathbb{R}$ have no interactions a priori (they are not even embedded in a common background space, unlike the case $Y=\partial X$) we find it difficult to choose a canonical interaction cost $c$.