The first solution is to use the official UI Automation API.

Microsoft UI Automation is an accessibility framework that enables Windows applications to provide and consume programmatic information about user interfaces (UIs). It provides programmatic access to most UI elements on the desktop. It enables assistive technology products, such as screen readers, to provide information about the UI to end users and to manipulate the UI by means other than standard input. UI Automation also allows automated test scripts to interact with the UI.

This code opens a c:\temp folder (or select one already opened) and adds a new tab to the Explorer window:

int main()
{
  CoInitialize(nullptr);

  // get a pidl for a given folder
  LPITEMIDLIST pidl;
  if (SUCCEEDED(SHParseDisplayName(L"c:\\temp", nullptr, &pidl, 0, nullptr)))
  {
    // open the folder (or activate an already matching opened one)
    SHOpenFolderAndSelectItems(pidl, 1, (LPCITEMIDLIST*)&pidl, 0);

    // find window that displays this pidl
    IShellWindows* windows;
    if (SUCCEEDED(CoCreateInstance(CLSID_ShellWindows, nullptr, CLSCTX_ALL, IID_PPV_ARGS(&windows))))
    {
      // build buffer from pidl
      VARIANT url;
      VariantInit(&url);
      if (SUCCEEDED(InitVariantFromBuffer(pidl, ILGetSize(pidl), &url))) // propvarutil.h
      {
        long hwnd = 0;
        IDispatch* disp;
        VARIANT empty;
        VariantInit(&empty);
        windows->FindWindowSW(&url, &empty, SWC_BROWSER, &hwnd, 0, &disp);
        VariantClear(&url);
        VariantClear(&empty); // useless but we always pair calls

        if (disp) // should be null but we never know...
        {
          disp->Release();
        }

        if (hwnd)
        {
          //needs UIAutomationCore.h & UIAutomationClient.h
          IUIAutomation* automation;
          if (SUCCEEDED(CoCreateInstance(CLSID_CUIAutomation8, nullptr, CLSCTX_ALL, IID_PPV_ARGS(&automation)))) // or CLSID_CUIAutomation
          {
            // get UIA element from handle
            IUIAutomationElement* window;
            automation->ElementFromHandle((UIA_HWND)(INT_PTR)hwnd, &window);
            if (window)
            {
              // window's class should be 'ShellTabWindowClass'
              // create inconditional "true" condition
              IUIAutomationCondition* trueCondition;
              automation->CreateTrueCondition(&trueCondition);
              if (trueCondition)
              {
                // create a tree walker to determine window's parent
                IUIAutomationTreeWalker* walker;
                automation->CreateTreeWalker(trueCondition, &walker);
                if (walker)
                {
                  IUIAutomationElement* parent;
                  walker->GetParentElement(window, &parent);
                  if (parent)
                  {
                    // create a condition to find the first button with AutomationId property set to AddButton
                    VARIANT v;
                    V_VT(&v) = VT_BSTR;
                    V_BSTR(&v) = SysAllocString(L"AddButton");
                    IUIAutomationCondition* condition;
                    automation->CreatePropertyCondition(UIA_AutomationIdPropertyId, v, &condition);
                    VariantClear(&v);
                    if (condition)
                    {
                      IUIAutomationElement* button;
                      // search button
                      parent->FindFirst(TreeScope_Subtree, condition, &button);
                      if (button)
                      {
                        // a button implements the "Invoke" pattern
                        IUnknown* unk;
                        button->GetCurrentPattern(UIA_InvokePatternId, &unk);
                        if (unk)
                        {
                          IUIAutomationInvokePattern* pattern;
                          unk->QueryInterface(&pattern);
                          if (pattern)
                          {
                            // press the button
                            pattern->Invoke();
                            pattern->Release();
                          }
                          unk->Release();
                        }
                        button->Release();
                      }
                      condition->Release();
                    }
                    parent->Release();
                  }
                  walker->Release();
                }
                trueCondition->Release();
              }
            }
            automation->Release();
          }
          CoUninitialize();
        }
      }
      windows->Release();
    }
    CoTaskMemFree(pidl);
  }
  CoUninitialize();
  return 0;
}

To be able to program UIA you generally first analyse the window's structure for example using the Inspect tool from Windows SDK or the newer Accessibility Insights to understand what actions you should execute. Here's an inspect screenshot on hightlighted Explorer's "Add new tab" button that tells us this button has an Automation Id property set to "AddButton":

PS: Sometimes, it's less obvious as Automation Id is not always set.

Another solution is to use the undocumented Explorer's WM_COMMAND 0xA21B (41499) which directly asks Explorer to open a new tab. So, with the previous example code, you can simply replace all UIA code by this, after you got ahold on the ShellTabWindowClass window:

...
if (hwnd)
{
  // window's class should be 'ShellTabWindowClass'
  // ask to open a new tab
  SendMessage((HWND)(INT_PTR)hwnd, WM_COMMAND, 0xA21B, 0);
}
...

Once you have opened a new tab, it's seen as a new window (view), so you can (re)start using CLSID_ShellWindows again and navigate this new view to any folder using something like what I have done here https://stackoverflow.com/a/78272475/403671 with Navigate2 call.