Here I Stand, although it doesn't have many similarities to the games you've already mentioned.

Struggle of Empires (or Conquest of the Empire) may be a good choice.

Other 4-6 players wargames could be Game of Thrones, Dust or Britannia.

There are some nice card-driven wargames but for 2-5 players: Sword of Rome (with the expansion) or Napoleonic Wars.

Conflict of Heroes: Awakening the Bear! - Russia 1941-1942 apparently plays well with 4 players, but does not play with more.

Shogun might be a good choice for you. It combines the feeling of combat and area control that Risk has with some of the more euro-style feelings you'll get from Puerto Rico and Settlers. It plays up to 5 players, and is a very good game.

Viktory II is the answer.

Also good is Nexus Ops, but only for 4 max.

Wizard Kings will cope with up to 7 players in the basic game, but you'd be advised to get some Wizard Kings - Expansion Maps to give you some room to move around!

Conquest of Paradise can have up to 4 players.

Attila the Hun: The Roman Empire in the East seats 6, and it's western sister game Alaric The Goth: Fall of the Western Roman Empire seats 7... But don't look so good!

Senji is a great Euro style war game that we had a chance to play GenCon 2008, it's best with 4-6 players, but can be played with 3 as well.

Another vote for Here I Stand. Not too hard to figure out, and lots of negotiation, backstabbing, etc.

I second Shogun.

You also might want to check out Diplomacy and Revolution: The Dutch Revolt 1568-1648.

Along the lines of the games you mentioned, there are Dust and Attack! / Attack! Expansion (soon to be republished together as Attack! Deluxe), all of which support up to 6 players. You might also consider Cosmic Encounter, which is scheduled for republication soon, although with only support for 5 players out of the box.

For games with a bit more bite, I agree with the suggestion of Here I Stand, although I really prefer playing with 3 players instead of 6 because there can be significant down time. I have not played Wizard Kings, but I think it's a good suggestion for a game to try with 6 players. Another to look at is Napoleon's Triumph, in which the multi-player game is handled by assigning leaders to the different players in a hierarchy for priority.

With 4-players, I would recommend Friedrich.

The game is played using an interesting 3-versus-1 game design, where the Prussian/Hanoverian player attempts to fend off all 3 of his/her rivals: Russia/Sweden, Austria/Imperial Army, & France. In game combat revolves around the play of tactical cards (that is, specially printed playing cards in 4 suits) and the hand management of these cards. If you are looking for a different type of wargame, then Friedrich might be the game for you and your group.

Conquest is a great 4-player game (it also has a 2-player version). It's becoming scarcer, with the designer passing away last year. Grand Conquest is an advanced version, adding siege weapons, additional troop types and terrain.

I'm surprised no one has mentioned Conquest of the Empire yet. Seats 6.

A personal favorite of mine is Succession Wars. Set in the Battletech Universe of 2025, you can play each of the 5 Great Houses of the Inner Sphere, plus a 6th player in control of ComStar (IIRC).

Another favorite, also sci-fi themed, is Stellar Conquest (for 4 players). Not too difficult to find, it's regularly offered up on eBay.

Among older games, there's 4-player goodness in Invasion: America, it's sister game Objective Moscow: the Death of Soviet Communism (both OOP and pricey SPI titles), and Invasion: America's illegitimate child, Fortress America (kin to Axis & Allies; prices fluctuate on this one).