The New Tetris uses smooth falling animation for tetrominoes.
The way this implementation mixes the falling animation with collision detection means that when a tetromino must be moved under an overhang to reach a ledge, by the time it has moved left one space with no support it has dropped more than zero spaces. In some cases it can therefore not reach a ledge.
Nothing can be done in the case of the O piece:
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There is no problem getting the O piece to here
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But if the ledge is further under the overhang we have a problem
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The piece can not move left until this point
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It cannot move left two spaces before it has dropped more than zero spaces
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There are, however, ways of getting a piece other than O further under the overhang by making use of "tactical rotation": rotating the piece in the right direction at the right time.
Examples
L/J shuffle
This is useful for completing spiral L monosquares on the left and spiral J monosquares on the right (in which case the move is reversed).
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Once the L is clear of the overhang it needs to slide two positions left to lock in place. This can't be done just by holding left
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Firm drop the L to hang off the edge
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Hold left, rotate clockwise then anticlockwise in quick succession
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The L is moving through the corner piece. Keep holding left and soft drop, so it slides left as soon as it can.
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Completed square
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If the space above the block is already occupied in such a way that the L cannot be overhung as in the second figure, a trickier version can be performed:
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Lower the L in this orientation
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Start to hold left
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Just as the L begins to move left, rotate anticlockwise
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The L is moving through the corner piece: keep holding left and soft drop it into place
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Finished square
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S/Z wiggle
This move is required for ZLLZ/SJJS and TTLZ/TTJS multisquares. The move is reversed for SJJS and TTJS multisquares.
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The space marked with  must be free
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That surely won't fit...
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Firm drop to here, then rotate clockwise then anticlockwise in quick succession
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The Z piece is sliding through this position. Firm drop it.
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Job done
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Reaching under overhangs
This works in a similar way to the trickier version of the L/J shuffle.
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Hold left from here
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As soon as the piece starts moving left, rotate anticlockwise (check if direction matters)
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Now use the L/J shuffle
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This can also be done with I pieces:
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Just holding left from here won't get the I piece in place
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Hold left from here
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As long as the gravity isn't too strong at this point in the game, you'll get to here (or a space further down)
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Rotate the I piece into place
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Building a T monosquare quickly
The quickest way to add the final piece of a T monosquare is to spin it into place:
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Firm drop the backwards-oriented T piece
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Rotate the piece twice quickly
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Finished square
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If there is no free space above:
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With the T piece oriented backwards hold left and soft drop
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It catches in the gap. Rotate the piece twice quickly
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Finished square
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