Can I increase chances of coop with friends in Dark Souls 3?

There's a few things controlling when you can summon someone (even with the password on).

First off, when you have password summoning on, the soul-range system is overridden, so your level 100 friend can come help your level 20 character, they will just get a normalizing damage reduction.

Second, if you are in an area where you have defeated all bosses in that area, you cannot summon anyone, regardless. You used to be able to forgoe this in Dark Souls and Dark Souls II with the Small White Sign Soapstone, but that doesn't seem to be the case in 3, as I haven't seen that item in-game. The exception to this rule is that you can do PVP summons (red sign soapstone, etc.).

Third, the summoner must have used an ember (or defeated a boss) restoring them to be a Host of Ember.

Finally, Dark Souls also enforces a "summoning cooldown" at times, limiting when you can summon your friend.

If all these conditions are met, you should be able to summon. Keep in mind it can take up to a few minutes (in my experience) for summoning signs to appear.

Also, just a sidenote: the Dark Souls netcode is sometimes a bit buggy. There are certain areas I cannot summon my friends in, but they can summon me, regardless of us having the same progress. While it's not foolproof, sometimes both the summoner and the summonee restarting Dark Souls 3 completely.

Sometimes it can be possible to summon a maximum of four friendly phantoms, although only certain areas allow this, you are able to. There are two items you need to do this: an ember, which allows you to summon and be invaded; the dried finger, which allows you to summon a third phantom. The dried finger can be purchased from the shrine handmaiden for a small amount of souls, and embers are found scattered throughout the game, and can sometimes be purchased from vendors in Firelink Shrine. Set a password for your friends, like, "tree" or something. It doesn't matter as long as you all have the same one. The password will allow players of any level to summon each other. Once you have two phantoms in your world use the dried finger and you will be able to summon a third phantom if the location allows for it. Be aware that this will allow two dark spirits to invade at once, but since it will be four versus two this should not be a problem.