It's hard to have much expectation of any advanced forms of life on a planet orbiting a star that once turned the planet into toast. The super smart ones would have left long before that happened, which will be our game plan, no doubt, if we are around 4 billion years from now.
Then there is the problem of young WDs having a surface temperature of around 150,000K. This smaller but much hotter object is about 35x more luminous than the Sun. Earth would need to have moved past Jupiter to maintain the same level of luminosity.
This exoplanet would then need to migrate inward as the WD cooled. But the WD cools very fast until it reaches, say, about ~ 6,000K where the next 500K drop in temp. takes over 1 billion years.
Because the size of a WD is tiny compared to the Sun, it would need to be around 60,000K to be as luminous (wattage) as the Sun for a planet at 1 AU. But, as mentioned, this high temperature drops quickly, so the HZ moves quickly as well. [My math could use some scrutiny, perhaps, since I'm still rushing around these days trying to finalize a move to a new house.]
As for locating the exoplanets of WDs, I would assume it is far less likely that transits would be seen given how small the disk are for WDs. But, radial spectroscopy is likely a little easier given their smaller mass.
The exoplanet.eu site shows 21 confirmed pulsars with exoplanets and 7 listed with WD or white dwarfs. The space.com article states, "Because sunlike stars are very common, and sunlike stars evolve into white dwarfs, there should also be a lot of planets around white dwarfs. And yet observations there have come up short, with only a handful of exotic examples. One is WD 0806-661b, a gas giant planet nearly eight times more massive than Jupiter that orbits at a distance of over 2,500 astronomical units, or 232.5 billion miles (373.7 billion kilometers), from its white dwarf star, meaning it takes more than 158,840 Earth years to complete one orbit. Another is PSR B1620-26 (AB) b, a gas giant that orbits a white dwarf-pulsar pair. There are two challenges for anyone interested in finding exoplanets around white dwarfs. One, they're very small and relatively dim, so the commonly used transit method, in which we stare at a star and wait for the exoplanet to cross in front of it, doesn't work. Two, white dwarfs don't have a lot of standout features in their spectra, so the other popular method, which involves watching the redshift and blueshift of spectral features as an orbiting planet tugs on its parent star, doesn't work either."
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