Neutrino oscillation experiments have so far provided the only evidence that neutrinos have mass. Since they measure the accumulated phase differences, they are only sensitive to the differences in masses, and thus never the absolute mass scale. A different measurement is necessary to probe the absolute mass scale. There are several experimental programs working on addressing this problem right now. While each is extremely challenging, there is a chance they may measure something.
But first an issue with your question. The electron neutrino doesn't have a mass. There is a neutrino called $\nu_1$ which has definite mass $m_1$ and so on for 2 and 3. The electron neutrino is the neutrino that interacts with an electron in a charged current interaction. It turns out that we know that $\nu_e$ is a linear combination of $\nu_1$, $\nu_2$, and $\nu_3$, and so on for $\nu_\mu$ and $\nu_\tau$. The matrix that describes how these guys mix is known as the PMNS matrix. So stating a mass for $\nu_e$ is quite misleading. In fact, it is even more confusing than that. Several experiments are sensitive to a mass term and only the electron neutrino is in play. But due to the kinematics of the experiments, the mass term isn't the same making a direct comparison impossible.
Finally, to answer your question, there are three* main classes of experiments. The theoretically simplest one is a tritium end point measurement. The state of the art as of today is KATRIN in Germany (there's a great photo of the experiment being moved through town). They measure the energy spectrum of electrons from tritium decay. At the very end of the spectrum there is a small feature due to the presence of the electron neutrino's effective mass. This program has the worst experimental constraints.
Next there is neutrinoless double beta decay. If lepton number is violated (neutrinos are Majorana) then if you wait for an atom to beta decay twice at the same time, it is possible that one neutrino will switch states and annihilate the other. If this happens two electrons with delta functions of energy will be emitted. The rate at which this happens scales with the effective mass of the electron neutrino (a different effective mass than before). The difficulties with these experiments are reducing the backgrounds. Still, there has been tremendous experimental progress lately and more will come in coming years.
The third main program is from astrophysics. The cosmic neutrino background (CNB) constitutes the second most abundant known particle in the universe after the CMB photons. They affect structure formation. At early times the neutrinos were relativistic, but as they cooled, given their masses, they have become non-relativistic (at least two of them anyway). This modifies structure formation. However, due to degeneracies in the astrophysical parameters, it is difficult to cleanly extract this number. Nonetheless, they have claimed the strongest bounds to date.
To put it all together, if one of these experiments measures a non-zero value of their parameter, this can be compared with the mixings and mass differences from oscillations to determine if one massless neutrino state can be ruled out or not.
*There is one other method to measure the absolute mass scale that could possibly succeed in coming years which is methods involving measuring the CNB directly such as PTOLEMY. Measuring the CNB is awesome in its own right. It would also determine if neutrinos are Dirac or Majorana, and would have sensitivity to the absolute mass scale. Three awesome goals means it is worth pursuing, but the experimental challenges are massive.