The obvious thing to do is make Austria, Savoy, or Tuscany your rival and wait for the Enemy Of My Enemy bonus to build up.
Hungary is his PU-minor, so unfortunately that doesn't work.
Qoff: I covered this in some detail, in this post:
How to further boost relations with PU-minor: via war.
Basically, you're going to have to go to war against Morocco, Poland, Savoy or Denmark - Hungary's Enemies.
If at all possible, you want it to be Savoy, because they are also Hungary's Rival.
That gives you an instant +10 "Declared War on Enemy" bonus, then there is another +20 available for Defender Our Territory and Liberated Our Province, totalling +30. Finally, if you annex one of Savoy's provinces, you can get another +10 for Conquered Rival Province. That one of course only applies if you DOW Savoy, who is the only nation who is both a Rival and an Enemy.
You need a total of +25 once you've maxed out Improve Relations, so you do need three out of four of those bonuses. So you have to do a bit of work to get in the right war, and have the right things happen in that war. See my post for more details.
In your case, the best strategy is to DOW Savoy, wait until they (or an ally) has some troops on Hungary's soil, then go kill them. Then win the war and annex one Savoy province. That will give you your +30. Note that the +10 Declared War on Enemy decays at -1 a year, from the
start of the war; so don't let the war go on more than 5 years!
EDIT: Oh, ninja'd by jdrou!
Can you use military access on a union partner?
No
Fight a war with their rival and take a province (+15).
That's great, I thought it was +10
Get into a war together and when it's over you should get +10 for "fought to the end".
Awesome, I forgot about that one too - it definitely applies to PU-minors? I'm sure it does.
So in that case, there's a total of
+70 available from all this:
- Declared War On Enemy, +10
- Conquered Rival Province, +15
- Defended Our Territory, +10
- Liberated Our Province, +25
- Fought To The End, +10
He only needs +25, so I wouldn't bother with Liberated Our Province which requires letting the enemy complete a siege against him.
And if he can't fight Savoy for some reason, he could DOW one of the other Enemies who isn't a Rival, foregoing Conquered Rival Province, but still getting +30 in total.
(I've also now edited my earlier post, that I linked to above, so it can be an accurate reference point for future; merging in the extra bonus jdrou mentioned, and fixing Conquered Rival to +15.)
EDIT 2: I just checked opinion_modifiers to double check the actual values. Liberated Our Province is actually +25! The others were all correct.