Combining compressed measurements from the same source
If you have $k$ individual measured vectors $\mathbf y$, each taken with an individual measurement matrix $\mathbf \Psi$, you actually have an interesting, third option:
$$\mathbf y = \begin{bmatrix} \mathbf y_1 \newline \vdots \newline \mathbf y_k \newline \vdots \newline \mathbf{y}_K \end{bmatrix} = \begin{bmatrix} \mathbf \Psi_1 \newline \vdots \newline \mathbf \Psi_k \newline \vdots \newline \mathbf \Psi_K \end{bmatrix} \mathbf x$$
So now you effectively have a (much) taller measurement matrix, so you are effectively "under-sampling less" and that will improve your estimate $\mathbf x$ based on the above equation. How much, I cannot recall an expression for off the top of my head.

Note also that you may have a tall measurement matrix now due to the stacking of the $\mathbf \Psi_k$'s. That is, you do not necessarily have an under-determined system anymore. This gives you the option of for example using least-squares estimation instead of $\ell_1$-norm optimisation. If you know that $\mathbf x$ is sparse, then it still makes sense to use sparse estimation ($\ell_1$-norm optimisation etc.), but if $\mathbf x$ is only approximately sparse, compressible then I would try least squares estimation as well.