Is HF the only method for trying to solve the many-electron Schrödinger equation?
So is this a correct assessment? Do we truly only use one method of trying to solve the many-electron Schrödinger equation? Is there any research being done on alternative methods (other than DFT)?


No, it is not correct. Saying that Hartree-Fock(HF) theory is the only method is falling short. Hartree-Fock is only one of many existing methods. While it is true that some single reference methods have a dependence on the HF solution (because they build upon it) when they include a reduced number of determinants, it is not the case for other. Particular mention deserves Coupled Cluster since it is insensible to the reference determinant found by HF. The proof is given by Thouless theorem (see Joshua Goings' post, or via The Internet Archive).
Thus, to give you an overview of the grand landscape of possible approaches I tried to compile the families of methods better established nowadays. I did not intend to specify individual methods because there are so many ... and people continue to investigate new ones. These are:

Deterministic


Semi-empirical (non ab initio): Hückel1/Tight binding2, Hubbard, ...
Wavefunction methods [Trygve Helgaker, Poul Jørgensen, Jeppe Olsen: 
Molecular Electronic‐Structure Theory. John Wiley and Sons, Ltd: 2000. DOI: 10.1002/9781119019572]


Hartree method. Caution: mathematicians use this name for a related method.
Hartree-Fock (HF): can be Restricted (RHF), Restricted Open (ROHF), Unrestricted (UHF) or Generalized (GHF).
Configuration Interaction (CI), up to order $N$, which is equivalent to Full-CI1/exact diagonalization2/direct variational approach3.
Multi-Configuration Self Consistent Field (MCSCF). [Chem. Rev. 2012, 112 (1), 108–181.]


Complete Active Space (CASSCF).
Two Configurations.
Manual selection of configurations or other.

Multi-Reference Configuration Interaction (MRCI)
CASPT2, NEVPT2, partially (PC) and strongly contracted (SC)
variants of the N-electron valence state second-order
perturbation theory
Multi-Reference Coupled Cluster methods (MRCC)
Many Body Perturbation Theory methods (MBPT) [Isaiah Shavitt, Rodney J. Bartlett: Many-Body Methods in Chemistry and Physics, MBPT and Coupled-Cluster Theory. Cambridge University Press: Cambridge, 2009. DOI: 10.1017/CBO9780511596834]


Coupled Cluster methods and its simplifications
Möller-Plesset perturbation theory of order $n$ (MPn)
Random Phase Approximation (RPA)
GW-Approximation

Explicit correlated methods R12/F12 [Chem. Rev. 2012, 112 (1), 75–107.]
Valence Bond Theory [Sason Shaik, Philippe C. Hiberty: A Chemist's Guide to Valence Bond Theory. John Wiley &amp; Sons, Inc.: 2008. DOI: 10.1002/9780470192597]

Density Functional Theory (DFT). [Robert G. Parr, Yang Weitao: Density-Functional Theory of Atoms and Molecules. Oxford University Press: USA, 1994. ISBN: 9780195092769]


Thomas-Fermi theory. Orbital-free methods. 
Kohn-Sham. Many density functional approximations developed, families are:


LSDA
GGA, hybrid GGA
MGGA, hybrid MGGA
OEP


Pair density functional theory [Comput. Theor. Chem. 2013, 1003, 91–96.]
Natural Orbital Functional Theory (NOFT) [S. Goedecker, C. J. Umrigar: Natural Orbital Functional Theory. In: J. Cioslowski (eds) Many-Electron Densities and Reduced Density Matrices. Mathematical and Computational Chemistry. Springer: 2000, Boston, MA. DOI: 10.1007/978-1-4615-4211-7_8]
Geminal Functional Theory [J. Chem. Phys. 2000, 112 (23), 10125–10130.]
k-particle density cummulants methods [Int. J. Quantum Chem. 2003, 95 (4-5), 404–423.]
Density Matrices methods


Density Matrix Renormalization Group (DMRG) [Annu. Rev. Phys. Chem. 2011, 62 (1), 465–481.]
Density Matrix Embedding Theory. Can be combined with several methods. [J. Chem. Theory Comput. 2016, 12 (6), 2706–2719.]
One-body Reduced Density Matrix Functional Theory (1-RDMFT) [J. Chem. Phys. 1985, 82 (12), 5604–5607.]
Variational 2-RDM Theory [Phys. Rev. Lett. 2012, 108 (26).]

Dynamical Mean Field Theory (DMFT). [J. Chem. Phys. 2011, 134 (9), 094115., arXiv:1012.3609 [cond-mat.str-el]] Starts with a Hubbard model of a lattice problem, then connects it to an impurity problem that can be solved with other methods.


Non-deterministic (stochastic)


Quantum Monte Carlo [Chem. Rev. 2012, 112 (1), 263–288.]


Variational Monte Carlo (VMC)
Fixed-Node Diffusion Monte Carlo (FNDMC).
LR-DMC, lattice-regularized diffusion Monte Carlo
Self-Healing Diffusion Monte Carlo
Green Function Monte Carlo (GFMC). GF2, self-consistent second-order Green’s function
Auxiliary Field Quantum Monte Carlo (AFQMC)
Reptation Quantum Monte Carlo
Full CI Quantum Monte Carlo (FCIQMC) [J. Chem. Phys. 2009, 131 (5), 054106.]
BDMC, bold diagrammatic Monte Carlo
Time-Dependent Quantum Monte Carlo



There are also mixtures of various methods.
The methods above have been tested with molecules by the quantum chemistry community but there are others that have been proven only for one dimension
or toy models. Operator methods are one example.

Many of them try to approximate the Full CI tensor that is described in a basis of determinant functions. It is impossible to treat such tensor for medium to large systems so the point is to take the minimum number of determinants that contribute significantly. The difference lies in how they choose those determinants. There are also other important distinctions among them, like if they are variational, perturbative or none, size-extensive, size-consistent, ...

In summary, each has its advantages and disadvantages. I recommend anyone interested to read
SOLVING THE SCHRÖDINGER EQUATION. Has Everything Been Tried? 2011, Imperial College Press. Ed.: Paul Popelier.

Furthermore, the analytic properties (read regularity) of the wavefunction can be exploited to devise new methods. As Harry Yseretant puts it in his book:


  Approximating the solutions is thus inordinately challenging, and it is conventionally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book tries to convince the reader that this conventional wisdom need not be ironclad: the regularity of the
  solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons.


Also, I would like to note that there is a collaborative project under the hood of the Simons Foundation whose purpose is precisely finding better methods to solve the electronic structure problem. And, of course, there are many more people working on the same task.

Footnotes


Naming adopted by the chemistry   community
Naming adopted by the physics     community
Naming adopted by the mathematics community


References


Mario Motta, David M. Ceperley, Garnet Kin-Lic Chan, John A. Gomez, Emanuel Gull, Sheng Guo, Carlos A. Jiménez-Hoyos, Tran Nguyen Lan, Jia Li et al. Towards the Solution of the Many-Electron Problem in Real Materials: Equation
of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods. Phys. Rev. X 2017, 7, 031059. DOI: 10.1103/PhysRevX.7.031059