One option is to have a lower-orbit set of satellites at 300km's so you can use the DP-10.
The other cheaper option is to right-click your parachutes and set the altitudes to deploy. Doesn't have to be high, but at least over 500m. Right before you enter the atmosphere, activate the chutes through staging and retract your antenna. You won't have any more control over the probe, however, with the parachutes activated, they will still deploy at the altitude you set them at. After it lands, recover it like normal.
If you really want a solar polar orbit, I'd probably start like you suggest, with a flyby over Jool's poles to change the inclination. You won't be able to do a full 90° inclination change just with a passive fly-by, but a burn at Jool periapsis (to maximize the Oberth effect) ought to be able to finish the job, as I've just suggested in an answer to the earlier question you linked to.
That said, if your actual goal is to build a communications network that will cover the entire solar system, there are other ways to do it that don't involve going far out of the ecliptic. For example, you could start by parking two high-power commsats in the same orbit around the Sun as Kerbin, but 120° ahead of and behind it. If you don't mind waiting a while, you can do this with very little delta-v; basically just enough to escape Kerbin's SOI, plus a tiny bit more to recircularize orbits once the satellites have drifted far enough from Kerbin.
Such a network will give you near-perfect coverage of Moho and Eve, and will also mostly cover the day sides of the outer planets, even when Kerbin itself happens to be behind the Sun. The triangle formed by the two commsats and Kerbin should be wide enough in the sky that eclipses should mostly be a non-issue, since even Ike isn't big enough when seen from Duna's surface to eclipse all three corners of the network at once. I suppose you might get occasional comms blackouts on Laythe when it goes behind Jool, though.
If you want to also cover the night sides of the outer planets, you'll have a bit more work to do. One option would be to send a constellation of three or four commsats into an orbit way out beyond Eeloo (again putting them in orbits about equally space around the Sun, in the corners of a triangle or a square, and with orbital periods matched as closely as you can). This will take considerably more delta-v than just drifting around Kerbin's orbit (and you'll need more antennas on the satellites, since they'll be further away), but probably still less than going into a polar orbit around the Sun.
The other option, of course, is to just deploy separate clusters of commsats around each planet. Honestly, this might be the simplest way to do it, especially if you can piggyback the satellites on another mission to the same planet.
(Also, by spamming enough max-level relay antennas on a single craft, it's actually not hard at all to make a commsat with more antenna power than the maxed-out DSN has. Launching one into, say, high polar Kerbin orbit way out beyond Minmus will improve your CommNet strength and also somewhat reduce the risk of random eclipses cutting off your comms at just the wrong moment.)
Best Answer
For most practical applications an easy solution is to have a ring of three evenly spaced satellites in high (2500km-7000km) circular equatorial orbits.
The only important thing is that the orbital periods of all three be the same, as the larger the difference, the sooner they will get out of position. Otherwise this scheme is resistant to placement errors.
Details:
Mun/Minmus occlusion: All the satellites can get occluded when either
This can be easily solved with the addition of one more satellite (at a later point), just make sure no two satellites line up when Kerbin blocks one of them.
(The "line up" will happen if, and only if, the phase angle differences between any one satellite, and any two other satellites adds up to somewhere near 180 degrees. So, for example, if we have three satellites 120 degrees apart, adding the fourth halfway between two of them is bad, as 120+60=180 and it can still get occluded. But adding it 40 degrees away is OK and will prevent occlusion from Mun/Minmus.)
Blind spots: Having a ring of satellites leaves blind spots on the surface (poles in case of equatorial ring). If the satellites are high up the spots will be really small, but any terrain irregularities close to them will easily block the signal. If you want probes at the poles, one solution is to just have another ring in a polar orbit. Or...
Rocket scientist alternative: if you are more of a rocket scientist than me, you can place four satellites in a tetrahedral formation around Kerbin. Such a formation will provide coverage on all of Kerbin's surface and cannot get fully occluded by Mun/Minmus. You can get more details in this KSP forum post, which takes it from this research paper (paywall).