It would prevent the spell from targeting you

Consider the wording of the amulet:

Whilst wearing this amulet you are hidden from [school] magic. You can't be targeted by such magic [...]

If we were to replace [school] with evocation, it'd be pretty clear by the description that you can't be targeted by evocation magic. I think it's equally clear in the case of divination.

Additionally, this amulet is of uncommon rarity. Blocking damage from a narrow range of spells is not unusually powerful for an uncommon magic item. A few examples of uncommon magic items that block spells:

• Brooch of Shielding: This gives resistance to force damage, and entirely negates Magic Missile (which is, in my experience, much more common than Mind Spike).
• Periapt of Health: This amulet blocks any spells that cause a disease in the target.
• Necklace of Adaptation: This necklace gives advantage to saving throws made against harmful gases such as Cloudkill and Stinking Cloud effects, as well as inhaled poison and some dragon breath weapons.

All of those effects are much stronger than Mind Spike.

There are no rules in D&D 5e that cover this.

Or to more explicit, there are overland travel rules, and there is nothing in them that change the pace at which you can travel overland based on your ability to move extra distance by taking a bonus action.

There is very little in the way of precedent however. Rogues can take a bonus action in combat to dash; this explicitly doesn't let them do it out of combat. Using your Action to Dash is presumably factored into overland movement (or trying harder and going faster).

So this ends up being left to the DM.

Why this ability to step as a bonus action might not make you move faster is that using magic might be distracting and tiring, much like using your action to dash is tiring. On the scale of combat, it isn't important (combat is presumed to be an intensive activity that wears you out), but in overland travel it might end up costing you more endurance than it gains you in speed.

On the other hand, even if it is effortful, common sense says that being able to teleport 30' in a fraction of a second every 6 seconds is going to make at least some travel much easier.

In terrain that isn't a completely flat, well maintained road, being able to "step" over stuff that might slow you down (a section of road with potholes, a stream, up a hill that would otherwise require climbing or a detour, past some brambles, etc) would speed you up.

Finally, there is the rule of cool. This character is an 18th level wizard who can teleport whenever she wants to. Being able to use this to increase your overland movement speed is both cool and creative. There isn't precedent that I am aware of that it shouldn't work, there is no huge balance impact on a wizard moving overland faster, and such an ability is well below the power scale of 18th level characters.

I'd say go for it. Maybe don't double travel speed, but grant the wizard +50% in the best situation (high quality roads), and halve the penalty of rough terrain (compared to the best situation) for the wizard.