Gift card coupons represent a stored credit value, therefore it is not possible to reuse these coupons to get the $10 off repeatedly with single coupon code.
You could send or provide a new unique gift card coupon worth $10 to anyone who signs up for the newsletter. You would have a single Gift card with multiple coupons. Their cound would be equal to the number of subscribers who also put more than $35 in their carts - but that minimum order amount may not be required.
As you can limit the gift card use for orders over $35, it would provide the same result.
I understand that the whole experience would be a bit different, though. You would need to generate the coupons (can be done using API) and send them to those who sign up for the newsletter. They won't see the discount until they add the received code to the shopping cart. On the other hand, you could also render this code dynamically in the site (e.g. in checkout) or even automatically add it to the shopping cart, if the order reaches $35 and the customer is subscribed... there are simply more options to handle that. The Gift cards are considered an alternative payment option and are applied to the total price (after the shipping and all taxes are applied).
Another option would be to customize the standard Order discounts - it is quite easy to get the flat order discounts back using approach described in this article - https://devnet.kentico.com/articles/fixed-amount-for-order-discount
This customization is not suitable for multi-currency setups, but if you use single currency, it should be fine. Please also note that unlike the Gift cards, the Order discounts are applied before tax and the calculation uses proportional distribution of the overall order discount to determine base price for each group of items with the same tax class. Order discounts also don't influence shipping.
Hope this helps!