After Multiple Sequence Alignment (MSA), data sets are produced on which models of evolution are used. I've seen in books that if there is strong similarity between the sequences then Maximum Parsimony (MP) is used, if that is low then Minimum evolution, and if there is very weak similarity then Distance based Method is used. However, all of these have their own cons. Since character based methods are more reliable (as book), Can these methods be used even when there is low sequence similarity? Are there any algorithm that builds the precise tree model without considering similarity of the sequences?
If you work with protein coding sequence and on the DNA level their to divergent, then it is better to use the amino acid sequence for alignment and phylogenetic reconstruction.
Better sampling can also help you, for this you can check the online databases with BLAST to find similar sequences that would fill the gap.
The problem with sequences with low sequence similarity is that it is difficult to identify homologous positions/characters and all alignment based methods rely on the assumption that the aligned positions are homologous.
Finally, there are alignment-free algorithms as well. (https://en.wikipedia.org/wiki/Alignment-free_sequence_analysis)
In general about algorithms for phylogenetics:
Nowadays, journals usually require you to run maximum likelihood (ML) analysis or Bayesian analysis. Both of these methods are referred to as model based. The most common programs used for these are RAxML (ML) and MrBayes (Bayesian). Another common software is MEGA (Molecular Evolutionary Genetics Analysis), which can run different tree building algorithms.
Usually, for quick reference maximum parsimony or neighbour-joining trees are good. For publication a maximum likelihood tree is very useful, since it returns a single best tree. While Bayesian analysis and maximum parsimony return a collection of trees.
It is also helpful different algorithms and if all return similar trees than you can be more sure about the relationships.