No, an Infectious Disease doctor will not send you to a LLMD. Very few ID doctors are actually willing to acknowledge long-term/chronic Lyme (i.e., very few are Lyme-Literate). You can get referrals to LLMD's through the ILADS website or through the Turn the Corner Foundation - http://ilads.org/ and http://turnthecorner.org - or you can post your location here and see if anyone knows of a good LLMD nearby.
Be aware also that most LLMD's do not take insurance because of the witch hunt against them that insurance coverage can increase their exposure to. Also, many insurance companies are not willing to pay for extended treatment for Lyme. And then there are a few lucky ones such as myself who has been able to get in with a LL Naturopath who is covered on my insurance, and all my treatment thus far has also been covered on my insurance ---but this is because of my other diagnoses I think (immune deficiency, etc.).
Chronic Lyme Disease, Chronic Bartonella (clinical dx only), Gluten & Sulfite Sensitivity, Many Food/Inhalant/Medication/Chemical Allergies & Intolerances, Asthma, Gut issues (dysmotility, non-specific inflammation), UCTD ("Secondary Lupus-Like Syndrome"), Osteoporosis, chronic Lymphopenia, etc.; G-Tube; Currently weaning off TPN.
Meds: Pulmicort, IV Doxycycline, Heparin (to flush PICC line), Claritin, Singulair, Domperidone, Colloidal Silver (topically & nasally), probiotics, Liver support herbs, Ailanthus, digestive enzymes, homeopathy.