Suppose $\phi:S^1\to S^1$ is a diffeomorphism. We can think of $D^2$ as the half sphere:
Actually this half sphere is just the rotation of the semicircle $C$ from $0$ to $\pi$ (I mean we rotate $S^1$ from $0$ to $\pi$ but we are only looking at the marks on the half sphere). So for any $\theta\in[0,\pi)$ and $p\in S^1$ we have a point $(\theta, p')\in D^2$, where $p'$ is the point corresponding to $p$ in $D^2$.
A bit of our extended diffeomorphism should look like $(0,p)\to(0,\phi(p))$.
So can it be something like$$ F(\theta,p) = \left\{ \begin{array}{ll} (\theta,\phi(p)) & \mbox{if }\text{$\phi(p)$ is on "our" side of the circle} \\ (\theta,-\phi(p)) & \mbox{if } \text{$\phi(p)$ is on "other" side of the circle} \end{array} \right.$$
Is this a diffeomorphism?
What you defined is (in general) not even a homeomorphism of hemisphere to itself (unless $\phi$ preserves the north and south poles).
On the other hand, there are several beautiful and very natural extension constructions for diffeomorphisms of the circle to diffeomorphisms of the disk, for instance, the Douady-Earle extension. Existence of a smooth extension (from sphere of dimension $k\le 2$ to ball of dimension $k+1$) was known long before them (combine Alexander's trick with the work by Rado, Kerikjarto and Moise, showing that in dimensions $\le 3$ TOP and DIFF categories coincide). However, such extensions (unlike DE) are noncanonical. (There is, however, a hard and canonical construction from $S^2$ to $B^3$ based on Beltrami equation and harmonic extensions of vector fields.)
More interestingly (and noncanonically) one can extend diffeomorphisms $S^k\to S^k$ to diffeomorphisms $B^{k+1}\to B^{k+1}$ for $k\le 5$. (As far as I know, the only way to prove this is to go through the equivalence of DIFF and PL in dimensions $\le 6$, due to Kirby and Siebenmann.) This stops working with $k=6$ and this is a part of the famous example of exotic 7d sphere by John Milnor, which changed the face of topology as we know it.