Bounded distance decoding beyond Babai
I realize that this is a very late reply, but the answer is yes. You can get an approximation factor of $2^{C n \log \log n/\log n}$ for any constant $C$ in polynomial time. In fact, you can do this with Babai's algorithm together with better preprocessing than LLL.

Unfortunately, I don't know of a great write up for this specific fact.

The main point is simply that the LLL algorithm is not our best polynomial-time algorithm for basis reduction. Instead, our best algorithms are generalizations of this, known as block reduction, or sometimes BKZ reduction (for Block Korkine-Zolotareff, though the name is not very helpful). The idea, originally due to Schnorr [1], is to look at various (projections of) $k$-dimensional sublattices and to find short bases for these (projections of) sublattices. LLL is just the case $k = 2$. I.e., LLL performs a bunch of simple operations on $2 \times 2$ matrices. See, e.g., [2] for a more recent version of this approach.

In polynomial time, we can take $k = C \cdot \log n$ for any constant $C &gt; 0$ (because we can solve more-or-less any lattice problem in exponential time), and the approximation factor that falls out is $2^{C' n \log \log n/\log n}$ for more-or-less any lattice problem that might interest you. As I said, I don't know of a specific write up about this for BDD, but the fact that we can get a $2^{C' n \log \log n/\log n}$-approximation to SVP via this approach was first proven by Ajtai, Kumar, and Sivakumar [3], as a consequence of their famous sieving algorithm. (See Corollary 11 in their paper.) You can then just use a generic reduction from BDD to SVP to get the same result for SVP (see, e.g., [4]).

You can also get this via BKZ reduction together with Babai's algorithm by observing that the approximation factor of Babai's algorithm is controlled by a certain quality measure on the basis---the decay of the Gram-Schmidt vectors---that we can bound for BKZ bases. The best write up that I know of about this property of Babai's algorithm is in my own lecture notes [5].

[1] Schnorr, C. P., A hierarchy of polynomial time lattice basis reduction algorithms, Theor. Comput. Sci. 53, 201-224 (1987). ZBL0642.10030..

[2] Gama, Nicolas; Nguyen, Phong Q., Finding short lattice vectors within Mordell’s inequality, STOC’08. Proceedings of the 40th annual ACM symposium on theory of computing 2008, Victoria, Canada, May 17--20, 2008. New York, NY: Association for Computing Machinery (ACM) (ISBN 978-1-60558-047-0). 207-216 (2008). ZBL1230.11153.

[3] Ajtai, Miklós; Kumar, Ravi; Sivakumar, D., A sieve algorithm for the shortest lattice vector problem, Proceedings of the thirty-third annual ACM symposium on theory of computing, STOC 2001. Hersonissos, Crete, Greece, July 6--8, 2001. New York, NY: ACM Press (ISBN 1-581-13349-9). 601-610 (2001). ZBL1323.68561.

[4] Lyubashevsky, Vadim; Micciancio, Daniele, On bounded distance decoding, unique shortest vectors, and the minimum distance problem, Halevi, Shai (ed.), Advances in cryptology -- CRYPTO 2009. 29th annual international cryptology conference, Santa Barbara, CA, USA, August 16--20, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-03355-1/pbk). Lecture Notes in Computer Science 5677, 577-594 (2009). ZBL1252.94084.

[5] http://www.noahsd.com/mini_lattices/05__babai.pdf .