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Hello from JustAnswer.
If someone develops an allergic rash while taking an antibiotic, it is generally correct to stop the antibiotic, but there should also be consideration of whether other steps should be taken
What is the condition for which you are taking the cephalexin?
How many days of the 10 day supply have you taken?
Can you attach a picture of the rash?
You are correct that the rash is faint, but it does look like it could be an early allergy rash. Consequently, it would be better to stop the cephalexin.
If the skin of the chemical burn from the deodorant has remained intact for a couple days, then it is less important to take an antibiotic, and it would be OK to use local cleaning to prevent infection. If there has been a loss of skin integrity, then it would be more important to take antibiotics. On the first day, the doctor would not know how a chemical burn will progress, but after a couple days, the severity of the burn would be more apparent. So, if there is a loss of skin integrity, then it would be better to change to an antibiotic in a different chemical class, such as Bactrim or doxycycline.
If I can provide any additional information, please let me know.
You are most welcome.