A Slim Jim antenna is similar to a J-Pole antenna, not only in terms of shape and size, but also technical characteristics: they share a very similar radiation pattern, with the main lobe pointing to the horizon and not the sky (assuming the antenna is physically pointing upwards). Even if you point it to the horizon, your “back” lobe will be exposed to a lot of noise from the ground (due to blackbody thermal radiation emitted by the Earth), so given your application, this is not an antenna you’re looking for.
Before proceeding to build a V-dipole (or any other antenna), can you inform me of your setup (from antenna to SDR)? Do you use an low-noise amplifier (LNA) at all?
How would I improve the impedance and phase matching?
When you electrically combine two antennas together via a coaxial cable (usually rated at 50 Ω), the impedance will change. This will increase your voltage standing wave ratio (VSWR), ultimately reducing the sensitivity of your setup. Even if you use a power combiner to deal with this issue, you will still need to make sure the antennas have a similar radiation pattern (i.e. pointing to the same direction) and the signals are added in-phase. To achieve in-phase addition, you may need to introduce an analog time delay before combining the signals together, by extending the length of one of the transmission lines (coaxial cables) appropriately:
(If the antennas receive the signals at the same time, then there is no need to introduce a phase delay.)
But again, I wouldn’t build a phased array for such application. I would just pick an LNA.