The Portuguese star recovered from the disappointment of a second round 75 that had left him one over at the halfway stage by birdieing the second and third at the stunning Seve Ballesteros-designed Porto Santo Golfe.
The 29 year old then added gains on the first two par threes, the fifth and seventh, and when he birdied the par five tenth - where he had taken seven on Friday - he was four under for the week and into a share of fifth.
That was three behind overnight leader Simon Thornton, with the Irishman about to tee off.
Also about to get underway were France's François Delamontagne, who was one off the lead at six under, and English duo Chris Gane and Jamie Elson on five under.
France's Alexandre Kaleka was the other big early mover on day three, holing his approach to the first for eagle and adding a birdie at the third to be three under through four and level with Lima, Michael Hoey, Alastair Forsyth and Andreas Hartø.
Lima double bogeyed the 15th but a closing birdie allowed him to sign for a 69 and set the clubhouse target at two under.
But he was no longer the leading Portuguese player in the field – Ricardo Santos was seven under for his round with three to play and looked likely to set a best-of-the-week score.
At six under for the week he was tied for third with Delamontagne and Gane, while Thornton was joined out in front by Hoey after the Northern Irishman completed an outward 33.