If "morning smoke" == milder, I can think of several that I happen to like with morning coffee. Various price ranges, too.

Macanudo Cafe: Possibly too mild if you like your coffee on the strong side, but pretty good with cream and sugar/Splenda.
Don Diego: I prefer one of the Babies with my second cup, black.
Casa Blanca: One of the few cigars I've found that are milder than Macanudos. Decaf, maybe?
Arturo Fuente Breva Royale or Curly Head, maybe a Hemingway Short Story if you're running late for the carpool.
Casa Torano: milder coffee, or cream/sugar.
Torano Master Blends 3: black coffee, medium or medium-dark roast.
Oliva Connecticut Reserve: Perhaps with a Starbucks Blonde Roast?
CAO Gold: Medium roast.
Casa de Garcia: Half-caf or a house brand.
Baccarat: Cream only, stir with head of cigar for sweetness.
Perdomo Lot 23: If you're grabbing coffee with a donut to go.