It honestly doesn't matter; even if we had perfect spectra of a terrestrial planet, we'd still struggle to figure what was (and wasn't) a signature of life.
And if we do detect something, noone will believe a signature from a terrestrial telescope; the preceise things we would be looking for are also the bands that the telescope has to look through in Earth's atmosphere.
But first you would need to detect the Earth-sized exoplanets orbiting the nearest Sun-like stars. TESS may detect the transiting ones, but the geometric probability for transits is not very high for planets in the habitable zone of Sun-like stars, thus you would not detect most of them with TESS. An astrometrical search with a precision of sub μas could find all these planets for the nearest Sun-like stars. SIM lite is/was such a project. In Europe there is a proposal for a mission called NEAT (Nearby Earth Astrometric Telescope) with 0.05 μas precision. It would search the 200 nearest Sun-like stars for Earth-sized planets. For more information please see http://neat.obs.ujf-grenoble.fr/NEAT.html .
And what capability would JWST offer in this area?
There's the "lunar hypertelescope" concept, too. Artemis might open an opportunity there.Image: prototype Southern Alps hypertelescope design from hypertelescope.org
