In Short and Long Term capital gains. If I purchase a crypto currency and then ADD to it a month later. Does the "start" date change to the date it was added?

You have two purchases and two items in inventory.  Here is an example of how it works.

• The first purchase of 1.0 unit of investment A is dated 01/01/2023 with a cost basis of $100.
• The second purchase of 1.0 unit of investment A is dated 02/01/2023 with a cost basis of $200.
• If you sell 1.5 units of investment A on 01/15/2024, you would report the sale of the first unit as a long term gain or loss because you held the investment for at least one year and one day.
• You would report the .5 unit as a short term gain or loss because you held the investment for less than one year.
• You are left with .5 unit of investment A in inventory with a cost basis of $100.

This example assumes that you are assigning inventory and costs under the FIFO or first in, first out method.  See also here.