  The main goal of this paper is to improve the result of Ostrovskii (2012) on
the finite determination of bilipschitz and coarse embeddability of locally
finite metric spaces into Banach spaces.
  There are two directions of the improvement:
  (1) Substantial decrease of distortion (from about $3000$ to $3+\ep$) is
achieved by replacing the barycentric gluing by the logarithmic spiral gluing.
This decrease in the distortion is particularly important when the finite
determination is applied to construction of embeddings.
  (2) Simplification of the proof: a collection of tricks employed in
Ostrovskii (2012) is no longer needed, in addition to the logarithmic spiral
gluing we use only the Brunel-Sucheston result on existence of spreading
models.
