It's not just Gamache. Supporting characters like Ruth Zardo, the bitter old poet, have arcs that span the entire series. Her first appearance in 'Still Life' is a caricature; by book ten, she's a profoundly complex figure. If you met her mid-arc, you'd either hate her without context or miss the tragedy of her development.
Penny treats every resident with that level of care. So yes, order matters immensely.