The I-129 can be used to change status from TN-1 to H-1B. It's one of the first questions on the form. The employer gets the approval back and you're in H-1B.
If your employer is unfamiliar with I-129s though it might be an idea for them to use an immigration lawyer, easy to make a mistake on that form, they have to include the approved labour certification for example.
Like I said above though my personal view is that H-1B is best used as a last resort. The problem with it is that it's a political hot potato and there are reams of all kinds of obscure regulations that restrict what you can and can't do, when it can be renewed, what the employer has to do, etc. A few examples that pop into my head for example are the advertising requirement for "H-1B dependent" employers, the requirement to notify USCIS of a layoff, the requirement to leave during a layoff (which is probably the case for TN-1 too but it's not as explicitly spelt out), the restrictions on renewal, the six-year maximum validity, the number of hours that must be worked, the quota on the number that can be issued and on and on.
And I absolutely guarantee Congress is going to add new requirements and regulations over the next few years.


