Hardwood to Tile Different Height Transition

That's a difficult one, and probably should have been thought through before installing the tile, not after.

I like using stone thresholds (commonly marble or granite) to terminate tile, but even a 6" wide would be sitting at a steep angle to make 3" vertical offset, and still be a stumbling hazard, at least. I'd probably take out that last/first tile and chew downwards until I could set it as a ramp of more reasonable slope.

If you are just going to have a 3" step and hope not to fall on it, you could make one from wood, or you might be able to set a 2" stone threshold on edge and a 4 or 6" one flat on top of it, filled with mortar to make the step; Or, remove wood in the threshold area until you can fit a solid 3" thick stone in there flat.

Alternatively, build up a ramp on the wooden floor side.

You say "adding a 4th step", so I presume that this new floor is at the top of a flight of 3 existing steps.

Since you've already laid down the tile, I'd suggest rebuilding the stairs to make each riser 1" taller. This will distribute the new height evenly over the steps so it will seem quite natural and unnoticed.

According to this site:

Stair riser heights The riser height shall be measured vertically between the leading edges of adjacent treads.

• Maximum stair step riser height (<= 7.75") or in some codes <=8"
• Minimum stair step riser height: >= 4"
• Step riser height uniformity <= 3/8" variation between successive step surfaces, or in some codes <= 3/16" / 4.8 mm.

So long as your total height is less than 23 1/4" (or 24" if you're in an 8" jurisdiction) you'll be good to go!

Of course, this isn't as easy as it may have been to have lowered the floor before tiling, but, this is where you are now. Of course, lowering the floor in a whole room may have been much more difficult than rebuilding a 3-step stair.