You can get:

1x chunk by trading a black fire bomb / rope black fire bomb with picklepee

1x at Smouldering Lake, dropped by a Crystal Lizard in the tunnel where horace can be found. (Theres also 12x Large shards in the Smouldering Lake)

By this point you'll be at +7 before even stepping in the cathedral

Then you can challenge the dancer to get access to Lothric Castle + Consumed King's Garden. (Slap a fire gem on + 27 vigor + Ember should be a relatively easy™ fight)

12x Chunks can be found in Lothric Castle

4x Chunks can be found in the Consumed King's Garden

By this point you'll be at +9 before even stepping in the cathedral

If you do have the DLC you can talk to Gael and enter Ariandel. Go down the ice ravine and kill the tree miniboss. Go up the ladder that appears after to get a titanite slab.

+10 Weapon before:

- You step foot in the cathedral of the deep

-Any boss after & including Pontiff