In many applications, it is not the matrix exponential $\mathrm{e}^A$ what is needed, but rather its action on some vector $B$, so there is literature dealing with the direct computation of the action of the matrix exponential: $$\mathrm{e}^A\times B,$$ that avoid the space and time complexities of storing and computing the full exponential. See for instance [1].
Are there similar methods for calculating the left action (for lack of a better name) of the exponential on a vector $B$: $$B\times \mathrm{e}^A?$$
[1] Al-Mohy, Awad H.; Higham, Nicholas J., Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput. 33, No. 2, 488-511 (2011). ZBL1234.65028. [1].
You can always use the fact that$$(B\times e^A)^T=(e^A)^T\times B^T=e^{A^T}\times B^T.$$