Is being recognized as Muslim one of the reasons for hijab?

As I understand, the primary reason for wearing hijab is because it's required in the Qur'an: > And tell the believing women... to wrap [a portion of] their headcovers over their chests and not expose their adornment except to... -- Qur'an 24:31 Other reasons include protection, modesty, and not being judged on beauty (ref. ). **Question**: Is being recognized as Muslim one of the reasons for hijab? This ayah partly supports this idea: > O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. **That is more suitable that they will be known and not be abused.** And ever is Allah Forgiving and Merciful. -- Qur'an 33:59 (see IslamAwakened for other translations) However, I could be misreading it (e.g., it could mean "recognized as women" rather than "recognized as Muslim", as [if I understand correctly] it was common for non-Muslims to observe veil at the time the ayah was revealed).