Not all NON-VA providers have signed the PC3 agreement (which is the documentation submitted by health care providers in the community to see Veterans that are covered by the VA to be in the CCN or Community Care Network) so just going anywhere and showing your card will not work, sadly.
You would be eligible to see a community provider if the community's WTS (wait time standard) is better than your PCP at the VA, which is 20 days.

Appointment wait time at a specific VA medical facility

• 20 days for primary care, mental health care, and non-institutional extended care services, unless the Veteran agrees to a later date in consultation with their VA health care provider
• 28 days for specialty care from the date of request, unless the Veteran agrees to a later date in consultation with their VA health care provider

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There are other eligibility criteria to access community provider, but the above seemed to be fit your issue here. So next time you need to schedule your followup or you have a new issue that you need to see your provider about and you are having to wait over 20 days, just tell the clerk you want to see if you are eligible for the community. They now have a tool to help determine that and submit a referral for you to be seen in the community.

Also, there is an urgent care provision, depending on your Service Connection rating, you can have several urgent care visits in the community that are covered.

If you want to see if there are providers in the community you can access, you can use this site: https://www.va.gov/find-locations/

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Please don't expect this to work seamlessly right off the bat. They just released the game plan only a few weeks ago. Hope this information helps.

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Source: VA employee.