One approach: Quantization
One approach is to quantize the values, then apply an existing LSH that works on discrete values. In other words, you split up the space $[0,1]$ into several ranges, map each real number to its containing range, and then apply an existing LSH that works with discrete values. This is essentially equivalent to first truncating each coordinate of the vector, then applying some discrete LSH.
For example, suppose you split the interval $[0,1]$ up into ten ranges, $[0,0.1)$, $[0.1,0.2)$, ..., $[0.8,0.9)$, and $[0.9,1.0]$. Now you can map any value in $[0,1]$ to its corresponding range -- this is basically just truncating to keep only the first digit in the decimal representation. Map each interval to a digit from 0 to 9: e..g, $[0,0.1) \mapsto 0$, etc. In this way, given any vector $v \in [0,1]^n$, you can get a new vector $w \in \{0,1,\dots,9\}^n$. For instance, the vector $v=(0.55,0.34,0.90,0.99) \in [0,1]^4$ maps to the new vector $w=(5,3,9,9) \in \{0,\dots,9\}^n$.
The new vector $w$ is in a discrete space, rather than in a continuous space. Now if you have a LSH that works on this discrete space, you can apply it to $w$. For instance, suppose $h_1:\{0,\dots,9\}^n \to \mathbb{R}$ is a locality-sensitive hash (LSH) on this discrete space. Let $f:[0,1]^n \to \{0,\dots,9\}^n$ be the mapping that sends each real number to its corresponding range, i.e., $f(v)=(g(v_1),\dots,g(v_n))$ where $g(x)= \lfloor 10x \rfloor$ is the truncation map $g:[0,1] \to \{0,\dots,9\}$. Define $h_0 : [0,1]^n \to \mathbb{R}$ by
$$h_0(v) = h_1(f(v)).$$
Then $h_0$ is a LSH on continuous values.
If you plan to generate multiple different LSHs, it's probably slightly better to apply the following tweak. Pick a random $n$-vector number $r \in [0,0.1)^n$. Now the LSH for $[0,1]^n$ is defined to be $h_0(v) = h_1(f(v+r))$ as above, where $f:[0,1]^n \to \{0,1,\dots,10\}^n$ is the truncation map defined as above and $h_1:\{0,1,\dots,10\}^n\to\mathbb{R}$ is a LSH on discrete values. This tweak lets you generate multiple independent LSH's for $[0,1]^n$. If you are only picking a single hash function, you can skip this step.
There's nothing special about base-10. Instead of using base-10, you can also use base-2 or any other base. The performance will depend upon the base you use as well as the specific discrete LSH you choose, so you might need to play with the options to see which is best overall. I suspect you will find that the optimal choice of base is base $b=2$ or $b=3$ (and, if your discrete LSH is based upon hashing a random subset of the coordinates, choosing a subset of size proportional to $\sqrt{n}/b$), but it's not possible to give a hard-and-fast rule: the optimum depends to some extent on the distribution of your data. That's why I suggest trying a few different parameter choices to see which seems to work best on your data.
This is basically the "truncation" idea in your original post (truncate all real values, then apply a discrete LSH), but now we apply it to the entire vector, rather than trying to split the vector in half. Of course, the discrete LSH might itself work by picking a few randomly chosen coordinates and hashing only those coordinates (e.g., using a universal hash).
Another approach: Use a LSH designed for continuous values
There are also some LSH's that are designed specifically for continuous real numbers. I don't know if there are any for the Euclidean distance.
A totally different approach: nearest-neighbor data structures
Finally, it's worth mentioning that there are also other approaches to this problem that don't involve LSH's at all. There are some data structures that are designed to support nearest-neighbor search (or approximate nearest-neighbor search). You might look at metric trees, k-d trees, and nearest-neighbor search.
However, beware of the curse of dimensionality. If the dimension $n$ is large, none of these approaches are likely to work well: high dimensions are just plain hard. For instance, one rule of thumb I've seen is that k-d trees tend not to work well when $n \ge 30$. I'm not sure any of these techniques will be terribly effective when $n$ is large: nearest-neighbor search in high dimensions is hard.
