Sorin Cosoveanu sent me a list of subregions of Eritrea which he believes to be current, as follows. He also supplied the information given in the Change history section for 2005 and 2006. He cites several Internet sources, but I wasn't able to access those pages.

Change history:

1993: I found two sources that agreed on the following list of secondary subdivisions of Eritrea. They didn't give a name for the type of division, so I call them districts, the default name for secondary subdivisions. In both cases they were presented as subdivisions of the provinces, which disappeared in a reorganization in 1996. Therefore, the first four letters of the HASC code indicate a province rather than a region of Eritrea. Also, according to both sources, Asmara (which was supposed to have province rank at that time) was contained in Debubawi Bahri. These districts may have disappeared along with the provinces in 1996. However, they offer a starting point for further research.

1. 1996-04-15: New constitution took effect. Eritrea reorganized into six regions; presumably the subregions were also formed then.
2. 2005: Adobha split from Karura; Asseb City split from Southern Denkalya; Hamelmalo split from Halhal; Mai-Aini formed from parts of Adi Keih, Adi Qwala, Dekemhare, Mendefera, Segeneiti, and Tserona; Molqi split from Shambuko; North Western Asmara split from North Eastern Asmara; Kudo Be'ur renamed Emni Haili; Omhajer renamed Goluj.
3. 2006: Central Denkalya and Southern Denkalya renamed Central Southern Red Sea and South Southern Red Sea, respectively.