How do I persist all querystring values in asp.net core 3.1 mvc?

In the Index action method, you could use ViewBag or ViewData to store the current search value (the CompanyName), then transfer it to the view page:

Then, for the sort links, you could use asp-route-{value} to add the current search value as the route parameter, and bind values via the ViewData or ViewBag. Code like this:

 <a asp-action="Index" asp-route-CompanyName="@ViewData["CurrentFilter"]" asp-route-sortOrder="@ViewData["CompanyNameSortParm"]">CompanyName</a>

The sample code as below:

Models:

public class Employee
{
    public int EmployeeId { get; set; }
    public string EmployeeName { get; set; } 
    public string CompanyName { get; set; }
     
}
public class Company
{
    public int CompanyId { get; set; }
    public string CompanyName { get; set; }
    public List<Employee> Employees { get; set; }
}
public class EmployeeVM
{
    public string CompanyName { get; set; }
    public List<Company> AllCompanies { get; set; }
    public List<Employee> Employees { get; set; }
}

Controller:

public class CompaniesController : Controller
{
    public IActionResult Index(string sortOrder, string CompanyName)
    { 
        ViewData["CompanyNameSortParm"] = String.IsNullOrEmpty(sortOrder) ? "companyname_desc" : "";
        ViewData["CurrentFilter"] = CompanyName;

        var employees = from s in repo.GetEmployees()
                       select s;
        if (!String.IsNullOrEmpty(CompanyName))
        {
            employees = employees.Where(s => s.CompanyName.Contains(CompanyName));
        }
        switch (sortOrder)
        {
            case "companyname_desc":
                employees = employees.OrderByDescending(s => s.CompanyName);
                break; 
            default:
                employees = employees.OrderBy(s => s.EmployeeId);
                break;
        }

        EmployeeVM vm = new EmployeeVM();
        vm.Employees = employees.ToList();
        vm.AllCompanies = repo.GetAllCompanies().ToList();

        return View(vm);
    }
}

Index Views:

@model netcore3.Data.EmployeeVM
 
<form asp-controller="Companies" asp-action="Index" method="get">
    <p>
        <label>Company Name: </label>
        <select asp-for="CompanyName" asp-items="@(new SelectList(Model.AllCompanies, "CompanyName","CompanyName"))">
            <option value="">All</option>
        </select>
        <input type="submit" value="Find" />
    </p>
</form>
<table class="table">
    <thead>
        <tr>
            <th>
                EmployeeName
            </th>
            <th>
                <a asp-action="Index" asp-route-CompanyName="@ViewData["CurrentFilter"]" asp-route-sortOrder="@ViewData["CompanyNameSortParm"]">CompanyName</a>
            </th>
            <th></th>
        </tr>
    </thead>
    <tbody>
        @foreach (var item in Model.Employees)
        {
            <tr>
                <td>
                    @Html.DisplayFor(modelItem => item.EmployeeName)
                </td>
                <td>
                    @Html.DisplayFor(modelItem => item.CompanyName)
                </td>
                <td>
                    <a asp-action="Edit" asp-route-id="@item.EmployeeId">Edit</a> |
                    <a asp-action="Details" asp-route-id="@item.EmployeeId">Details</a> |
                    <a asp-action="Delete" asp-route-id="@item.EmployeeId">Delete</a>
                </td>
            </tr>
        }
    </tbody>
</table>

The result like this:

Reference:

Tutorial: Add sorting, filtering, and paging - ASP.NET MVC with EF Core

Anchor Tag Helper in ASP.NET Core