Even though your school district doesn't have any gifted programs at this time, they might offer testing for gifted-id (please know, I'm not in NY so I have absolutely *no* knowledge of your local area - just offering up suggestions!). If you can get testing through the school district it will be at no charge, so I would at least look into it. Did you ask the teacher specifically about testing, or about whether or not there are any programs within the school district that your dd might be able to participate in if she's id'd as gifted? Did you ask her for any advice re what she recommends for your dd?
FWIW, programs offered outside of school have been *so* beneficial to my kids. If you have programs in your area that you think your dd would like to participate in, I'd have her tested - but I'd also first look into what specifically is required in terms of testing for the programs you're interested in. This is just a for instance, but we've participated in some of the programs offered through Johns Hopkins CTY. With CTY, you can sign your elementary-age child up for their talent search testing even if your child hasn't had any type of formal IQ/achievement testing (hopefully my advice isn't out-of-date, but we did this for our youngest dd when she still hadn't been through any testing). Once your child is signed up, CTY has some programs that I think all families can attend, and others that your child has to qualify for through the CTY testing - but the CTY testing is relatively inexpensive compared to private IQ testing. It doesn't give you test data that you'll be able to use in advocating elsewhere, but it's a relatively inexpensive way into the program for a student that qualifies.
You didn't mention how old your dd is - I am guessing early elementary - if she's older I might have a few other ideas

Best wishes,
polarbear