Explicit isomorphism between $L^\infty[0,1]$ and its hyperplanes

Question

The Banach space $L^\infty[0,1]$ is isomorphic to all its hyperplanes (closed linear subspaces of codimension one). One way of seeing this is the following:

1. all hyperplanes of a Banach space are mutually isomorphic
2. $\ell_\infty$ is isomorphic to its hyperplane $\{x\in\ell_\infty\colon x(1)=0\}$.
3. $L^\infty[0,1]$ is isomorphic to $\ell_\infty$.

In the above list the step (2) can be made explicit. On the other hand, the third item (3) cannot; see this question and its answer by Robert Israel.

Is there a direct way of proving the above statement which allows for an explicit construction?

In particular, I would be interested in an answer to the question of whether it is possible to construct an explicit isomorphism between $L^\infty[0,1]$ and its hyperplane $\{f\in L^\infty[0,1]\colon \int_{0}^{1}f(t)\,\mathrm{d}t=0\}$. This question is related to this question on mathoverlow.