Some new information came up during a research.
In November 1957 the cargo motorship PROFITIS ILIAS was registered in Piraeus as "under reconstruction in Perama". She was listed as: 125 grt, 29.86 x 5.22 m. The Piraeus Register mentions that she was "formerly named GAVRIEL under the flag of Israel, register number 37, and previously named DIMITRIOS."
DIMITRIOS was the last name of TETI F. and under that name she was abandoned and half sank in Haifa, being deleted from the Greek Register at the end of 1953. It is probable that DIMITRIOS (ex-TETI F.) is the same ship as PROFITIS ILIAS (ex-GAVRIEL, ex-DIMITRIOS.)
If someone can provide more information on the "number 37" of the Israeli register, it could help to verify this story.
PROFITIS ILIAS was sold again in 1957 and renamed ELENI I, but her final history was not recorded.