Buy more bank space. Ask any banker to Expand my vault, or press +40 slots in the bank window's footer (or the + after the last tab). Each expansion adds 40 slots, half a tab, and two of them open a whole new tab, up to 800 slots across 10 tabs. Everyone keeps the 480 slots they have now.
Prices climb as you go: 5,000, 12,000, 25,000, 50,000, 100,000, 200,000, 400,000 and finally 800,000 coins. The confirm shows this price and the next one before you pay.
Pay from your pouch or your bank. Coins come out of your coin pouch first, then from coins stored in your bank, so you never have to withdraw them first.
Locked slots in a half-open tab show a padlock: click one to see the price of the next expansion.
Changed
With more than 7 bank tabs, the tabs you are not on show only their icon, so all of them fit. Hover a tab for its name. The new tabs start as Vault, Hoard, Spare and Deep and can be renamed like the others.