In the torn paper channel, a transmitted codeword is broken at random locations into fragments that arrive at the decoder in an unordered manner. A central theoretical challenge within this model is global alignment -- the task of determining each fragment's original position -- in order to faithfully reconstruct the entire codeword. Prior work by Shomorony and Vahid introduced an interleaved-pilot scheme that successfully achieved a vanishing error probability. However, their alignment strategy relies heavily on global statistics, requiring fragments to exceed a minimum length and effectively discarding many shorter ones as erasures, which results in rates significantly below capacity. To address this gap, we propose an improved coding scheme that achieves a provable rate increase through a novel approach we call \textit{local alignment}. This approach identifies global alignment bits within each fragment using only local information, allowing the decoder to determine the positions of fragments that are shorter than those used in previous work. Consequently, the decoder can extract information from a much larger fraction of the channel output than in previous work, yielding significantly higher rates. Furthermore, we extend our analysis to torn paper coding with lost pieces (TPC-LP), a generalized model that accounts for length-dependent fragment deletion. For a class of TPC-LP channels that delete all fragments below a logarithmic length threshold while allowing arbitrary length-dependent deletion probabilities for longer fragments, we show that the proposed local alignment strategy achieves an arbitrarily small additive gap to capacity as the threshold increases.