how to use database name as a value

Question

I have more than 1000 access databases with the same table names.Each database has a unique name. In each database, I want to add column to one existing table and populate that new column (all rows in the table would be the same) with the database name. I found the example below - How can I alter this to give the new column the name of the database.

ALTER TABLE TestTable ADD NewColumn INT;

UPDATE TestTable SET NewColumn = 0;