In order to unlock time trial stages, you need to beat certain chapters in the game, or beat the one-star qualifying time on the following stages:

Time Trial Stage	How to unlock
Arland	Complete Chapter One in Story Mode
Atrium 1	Beat a qualifying time on Convoy 2
Atrium 2	Beat a qualifying time on Atrium 1
Burfield	Beat a qualifying time on Heat
Chase	Complete Chapter Two in Story Mode
Convoy 1	Complete Chapter Eight in Story Mode
Convoy 2	Beat a qualifying time on Convoy 1
Cranes 1	Beat a qualifying time on Burfield
Cranes 2	Complete Chapter Four in Story Mode
Edge	Complete the Prologue in Story Mode
Factory	Complete Chapter Six in Story Mode
Flight	Beat a qualifying time on Arland
Heat	Complete Chapter Three in Story Mode
New Eden	Complete Chapter Five in Story Mode
Office	Complete Chapter Seven in Story Mode
Playground 1	Complete "Training Area" in Story Mode
Playground 2	Beat a qualifying time on Playground 1
Playground 3	Beat a qualifying time on Playground 2
Shard 1	Complete Chapter Nine in Story Mode
Shard 2	Beat a qualifying time on Shard 1
Stormdrains 1	Beat a qualifying time on Chase
Stormdrains 2	Beat a qualifying time on Stormdrain 1
Stormdrains 3	Beat a qualifying time on Stormdrain 2