Currently there is no ability to change the order of the fields/columns in the contact table. The custom fields are at the end and in a somewhat random order. For example: we have a custom field for "preferred pronouns", which is the one of the last fields on our table. It is listed after address, phone, email, etc. This doesn't make sense, and it doesn't flow well for a form.
It would also be helpful to adjust the width of the columns for this table. It looks messy and involves too much scrolling.