As a result of research on how the algorithm of multiplication was taught in the early 1900's, I found that part of one particular process could be extracted to teach the multiplication facts. Let's take a look at the prerequisites and the "new" but old idea of teaching multiplication facts without depending entirely on rote memory.
Before beginning two skills should have been taught. Ideally, students can add and subtract by the time their ready to learn multiplication facts. Also, they need to know how to add and subtract 2-digit by 2-digit, as well as borrow or regroup.

After the prerequisites are met, vocabulary and the scale which makes it all work can be introduced. Teach the basic concept of multiplication. Then, teach the definitions of multiplicand, multiplier, product, commutative property of multiplication, and the timetables for 1, 2, 5, and 10 in the manner you prefer. I have articles on learning these timetables. The links for them are at the end of this article.

Once those previous mentioned facts are taught, the following scale is to be introduced.

SCALE:
1 = 1
2 = 2
3 = 5 – 2
4 = 5 – 1
5 = 5
6 = 5 + 1
7 = 5+ 2
8 = 10 – 2
9 = 10 - 1

The first column of the scale shows the digits of the multiplier. The second column shows their equivalents. For example, the scale shows 7 = 5 + 2. To multiply a number by 7 you multiply by the number by 5 and by 2.

Now, let's apply the scale to one of the harder facts to remember such as 6 x 7.
The scale shows the multiplier, 7, converts to "5 + 2."

So, you would
1) Multiply the multiplicand, 6, by 5 to get 30
2) Multiply the multiplicand, 6, by 2 to get 12
3) Add the two products, 30 and 12 to get 42
In summary:
6 x 7 = (6 x 5) + (6 x 2) = 42
6 x 5 = 30
6 x 2 = 12
Thus, 30 + 12 = 42

Another example:
Let's apply the scale to one of the harder facts to remember such as 6 x 8.
The scale shows the multiplier, 8, converts to "10 – 2."

So you would
1) Multiply the multiplicand, 6, by 10 to get 60
2) Multiply the multiplicand, 6, by 2 to get 12
3) Subtract the two products, 60 and 12 to get 48

In summary:
6 x 8 = (6 x 10) - (6 x 2) = 48
6 x 10 = 80
6 x 2 = 12
Thus, 80 - 12 = 48

In essence, the students only need to know the 1, 2, 5 and 10 multiplication facts in order to learn the remaining 3, 4, 6, 7, 8, and 9 timetables. However, I find this method most useful for learning the 6, 7, 8 multiplication facts. Overall, this method can work for all ages especially for remediation of older school-aged students and adults. It can serve as another tool to compute the answer with a higher degree of accuracy.

The name of the book referenced is New Method in Multiplication and Division, by William T. Call, published in 1913. A reprint is available in hardback and paperback.