So, there's the daily mission from the Ferengi with the asteroid near DS9.

Where else can I mine for dilithium other than receiving dilithium through missions and duty assignments?

"Mine" in the sense of actual mining...I don't think there's any other place where you can do that.
But there's the "Academy Event" where you can collect Particles you can then trade for Dilithium:

http://www.stowiki.org/Mission:_Academy_Event

Those are your options. You have The sattelite repair, Dili mining. For dailys, earns about 800-1000 dili. The star fleet one gives you the ablity to earn from 460 -1400 give or take.

The deferi dailies still give dilithium too

It make you feel like a real farmer playing that annoying mining game.

Here's the the stuff in the game that provides dilithium that are shared by factions.

Faction shared (Federation and KDF)
Investigate Officer Reports: Still gives 1440 daily, just have to play a little longer. (not active right now)
Academy event: 1500-3000 seems to be the average range people can get, but the wiki says up to 5000 with luck. Sometimes available twice a day.
Lore missions: 480 daily for getting the right answer. Wiki has lore section for shortcut.
Borg Invasion: 480 daily
Exploration mission daily: 1440 for 3 missions in a cluster. Diplomatic investigation fills one out fast.
Aiding Deferi: 1440 daily, unlocks after beating breen episodes. Rescue and Emancipation unlock too this way.
Rescue captives: 480, daily. all you have to really do is rescue the Deferi from a Breen Patrol. Defeating the Breen is optional for Feds, a must for klingons.
Emancipation: Freeing Deferi slaves from Breen. 480 daily.
Asteroid mining: variable outcome, but tends to be several hundred. Daily.
Nukara prime: 640 for zone wides, daily. Thinning the Hive requires 250 kills for it to count.
Traelus system repair: 480 daily, fighting at the end of mission.
Sh'mar Distress call: 480 daily.
Defense contract: 480 daily, bonus of deuterium reserves at the end. Computer core fragments drop for Salvage dispute.
Salvage Dispute: 1440, requires 10 computer core fragments. Repeatable.
Big Dig Daily fleet action: 1440
Breaking the planet: 1440

Doff assignments
Contraband: 2000
relocate colonists: 500
transport settlers: 500
grant passage to colonists: 500
most doff assignments give a small amount of dilithium as reward for completing it.

Federation has 7 PvP missons that reward 1440 (daily?), Klingons have 6 that grant 1440, plus numerous ones that reward 480.

As you can see, there are plenty of ways of gaining dilthium still in this game. All it takes is a little bit of time to get it. That's why we have the system like we do.

List of Available SOLO Dailies for Dil on Fed Side:

Academy Quiz: 480
Investigate Officer Reports (when they fix it): 1440
Explore Strange New Worlds: 1440
Traelus Satellite Repair: 480
Shmar Defense: 480
Alhena Defense Contract: 480
Deferi Daily Defense: 1440
Emancipation (Zaria system, orellius block): 480
Breen Patrol: 480
Dilithium Mining: 600

Additional Sources (NOT Solo play):

Borg Sector Incursions: 480 each once per day
Fleet Actions: 1440 each ONCE per day, 480 per every run
Ker'rat Dailies: 1440 each (there are 2 of them I think)
Any Number of others I may or may not have forgotten: 480 to 1440 each.
DOff Missions: 5-2000 depending on mission

For Klingons:

List is too damn long. If you're having trouble getting dil as a klink, you need a new brain.

I mine the asteroid some if I'm in the DS9 area. If not I usually hit the "Duty Officers" and "Explore the Clusters." By the time I do those, and my regular missions. I don't have much time to hit the other missions to farm.

Everyone else forgot to mention it, So i will...
*Your Wallet*

I believe the Vault shuttle mission awarded Dil when it wasn't event time, but that might have changed after S7.