Theoretically a CT resident could have purchased a handgun using an Eligibility Certificate, then applied for a pistol permit at a later date. A temporary permit would then allow the person to carry the handgun legally. A alternative would be to use a handgun owned by another lawfully licensed person.
Here's the text of an e-mail I received from CT DPS in response to my inquiry:
A temporary permit allows a person to carry but they cannot purchase.
Sergeant Douglas A. Hall
Connecticut State Police
Special Licensing and Firearms Unit
1111 Country Club Road
Middletown, CT 06457
Tele 860-685-8003
Fax 860-685-8644
P.S. Sgt. Hall is an attorney